Warwick Economics Research Paper Series
Voting in Legislative Elections Under
Plurality Rule
Niall Hughes
December, 2015
Series Number: 1097
ISSN 2059-4283 (online)
ISSN 0083-7350 (print)
This paper also appears as CRETA Discussion Paper No: 3
VOTING IN LEGISLATIVE ELECTIONS UNDER
PLURALITY RULE∗
Niall Hughes
University of Warwick
n.e.hughes@warwick.ac.uk
June 2015
Abstract
Models of single district plurality elections show that with three parties anything
can happen - extreme policies can win regardless of voter preferences. I show that
when single district elections are used to fill a legislature we get back to a world where
the median voter matters. An extreme policy will generally only come about if it is
preferred to a more moderate policy by the median voter in a majority of districts. The
mere existence of a centrist party can lead to moderate outcomes even if the party itself
wins few seats. Furthermore, I show that while standard single district elections always
have misaligned voting i.e. some voters do not vote for their preferred choice, equilibria
of the legislative election exist with no misaligned voting in any district. Finally, I
show that when parties are impatient, a fixed rule on how legislative bargaining occurs
will lead to more coalition governments, while uncertainty will favour single party
governments.
Keywords: Strategic Voting, Legislative Elections, Duverger’s Law, Plurality Rule,
Polarization, Poisson Games.
JEL Classification Number: C71, C72, D71, D72, D78.
∗
I would like to thank Massimo Morelli, Richard Van Weelden, David K. Levine, Gerard Roland, Andrea
Mattozzi, Salvatore Nunnari, Helios Herrera, Nolan McCarty as well as seminar participants at the Columbia
Political Economy Breakfast, EPSA General Conference 2012, and IMT Lucca for many useful comments
and suggestions.
1
1
Introduction
Plurality rule (a.k.a. first-past-the-post) is used to elect legislatures in the U.S., U.K.,
Canada, India, Pakistan, Malaysia as well as a host of other former British colonies - yet
we know very little about how it performs in such settings. The literature on single district
elections shows that plurality rule performs well when there are only two candidates but
poorly when there are more.1 Indeed, plurality has recently been deemed the worst voting
rule by a panel of voting theorists.2 However, the objectives of voters differ in single district
and legislative elections. In a legislative election, many districts hold simultaneous plurality
elections and the winner of each district takes a seat in a legislature. Once all seats are filled,
the elected politicians bargain over the formation of government and implement policy. If
voters only care about which policy is implemented in the legislature, they will cast their
ballots to influence the outcome of the legislative bargaining stage. A voter’s preferred
candidate will therefore depend on the results in other districts. In contrast, in a single
district election - such as a mayoral election - a voter’s preference ordering over candidates
is fixed, as only the local result matters. These different objectives are at the heart of this
paper. I show that when three parties compete for legislative seats and voters care about
national policy, several undesirable properties of plurality rule are mitigated.
While there has been some key work on voting strategies in legislative elections under
proportional representation (PR), notably Austen-Smith and Banks (1988) and Baron and
Diermeier (2001), there has been scant attention paid to the question of how voters should
act when three parties compete in a legislative election under plurality rule. Studies of
plurality rule have either focused on two-party legislative competition or else on three-party
single district elections, in which voters only care about the result in that district. In the
former case, as voters face a choice of two parties, they have no strategic decision to make they simply vote for their favourite. However, for almost all countries using plurality rule,
with the notable exception of the U.S., politics is not a two-party game: the U.K. has the
Conservatives, Labour and the Liberal Democrats; Canada has the Conservatives, Liberals,
and New Democrats; India has Congress, BJP and many smaller parties.3 With a choice
of three candidates, voters must consider how others will vote when deciding on their own
ballot choice.
In a single plurality election, only one candidate can win. Therefore, when faced with
1
See for example Myerson (2000) and Myerson (2002).
See Laslier (2012)
3
The recent 2015 UK election witnessed a further fragmentation of the political landscape with the
Scottish National Party gaining a large number of seats. Other countries with plurality rule and multiple
parties represented in the legislature include: Bangladesh, Botswana, Kenya, Liberia, Malawi, Malaysia,
Mongolia, Pakistan, Trinidad & Tobago, Tanzania, Uganda, and Zambia.
2
2
a choice of three options, voters who prefer the candidate expected to come third have an
incentive to abandon him and instead vote for their second favourite, so that in equilibrium
only two candidates receive votes. These are the only serious candidates. This effect, known
as Duverger’s law4 , was first stated by Henry Droop in 1869:
“Each elector has practically only a choice between two candidates or sets of
candidates. As success depends upon obtaining a majority of the aggregate votes
of all the electors, an election is usually reduced to a contest between the two
most popular candidates or sets of candidates. Even if other candidates go to
the poll, the electors usually find out that their votes will be thrown away, unless
given in favour of one or other of the parties between whom the election really
lies.” (Droop cited in Riker (1982), p. 756)
A vast literature has pointed out two implications of Duverger’s law in single district
elections.5 First, “anything goes”: the equilibrium is completely driven by voters’ beliefs,
so any of the three candidates could be abandoned, leaving the other two to share the
vote. This means that, regardless of voter preferences, there can always be polarisation where a race between the two extreme choices results in an implemented policy far away
from the centre. Second, when each of the three choices is preferred by some voter, there
will always be misaligned voting. That is, some people will vote for an option which is not
their most preferred. Misaligned voting undermines the legitimacy of the elected candidate:
one candidate may win a majority simply to “keep out” a more despised opponent, so the
winner’s policies may actually be preferred by relatively few voters.
In this paper, I model a legislative election in which each voter casts a ballot in a local
district but their utility depends on policy determined in the national parliament. I show that
while polarisation and misaligned voting are always possible in stand-alone multi-candidate
plurality elections, they can both be mitigated in a legislative election setting.
The intuition for the polarisation result is as follows. For any party to win a majority of
seats in the legislature it must be that they are preferred to some alternative by a majority of
voters in a majority of districts. I show that the alternative to a left majority will generally
not be a right majority but rather a moderate coalition government. Therefore, for an
extreme policy to come about, it must be that the median voter in the median district
prefers this policy to the moderate coalition policy. This contrasts with single plurality
elections where extreme policy outcomes are always possible, regardless of preferences.
4
The law takes it’s name from French sociologist Maurice Duverger who popularised the idea in his book
Political Parties. While Riker (1982) argues that Duverger’s law should be interpreted as the tendency of
plurality rule to bring about a two-party system, most scholars use the term to describe the local effect: in
any one district only two candidates will receive votes. I also use the local interpretation.
5
See Palfrey (1989), Myerson and Weber (1993), Cox (1997), Fey (1997), Myerson (2002), Myatt (2007).
3
The misaligned voting result stems from the fact that voters condition their ballots on a
wider set of events in my setting. In a standard plurality election, voters condition their vote
on the likelihood of being pivotal in their district. However, in a legislative election, voters
will condition their ballot choice on their vote being pivotal and their district being decisive
in determining the government policy. In many cases a district will be decisive between two
policies, even though there are three candidates. For example, a district might be decisive
in either granting a majority of seats to a non-centrist party, say the left party, or bringing
about a coalition by electing one of the other parties. Under many bargaining rules this
coalition policy will be the same regardless of which of the weaker parties is elected. So,
voters only face a choice between two policies: that of the left party and that of the coalition.
When voters have a choice over two policies there can be no misaligned voting - everyone
must be voting for their preferred option of the two.
One technical contribution of the paper is to extend the Poisson games framework of
Myerson (2000) to a multi-district setting. I show in Lemma 1 that the Magnitude Theorem
and it’s Corollary can be used to rank the likelihood of various pivotal events across districts.
This greatly reduces the complexity of working with multi-district pivotal events and makes
the problem much more tractable.
I examine the workings of my model under several legislative bargaining settings - varying
the scope of bargaining, the patience of politicians, and the bargaining protocol. Government
formation processes do vary across countries. In some countries potential coalition partners
may bargain jointly over policy and perks, while in others perks may be insufficient to
overcome ideological differences. The patience of politicians may also differ across countries
depending on aspects such as how quickly successive rounds of bargaining occur, and how
likely politicians are to be re-elected.6 A further feature of government formation which has
been studied extensively is the protocol for selecting a formateur (a.k.a. proposer). The two
standard cases are random recognition - where a party’s probability of being the formateur
in each round of bargaining is equal to its seat share - and fixed order - where the largest
party makes the first offer, then the second largest, and so on. Diermeier and Merlo (2004)
analyse 313 government formations in Western European over the period 1945–1997 and find
the data favours random recognition rule.7 On the other hand, Bandyopadhyay, Chatterjee,
and Sjöström (2011) note that a fixed order of bargaining is constitutionally enshrined in
Greece and Bulgaria, and is a strong norm in the U.K. and India, where elections are held
under plurality rule. Therefore, I examine these various combinations to see how robust the
6
See Austen-Smith and Banks (2005) and Banks and Duggan (2006) for a discussion of discount rates
in legislative bargaining.
7
For a critique of the empirical support for random recognition see Laver, Marchi, and Mutlu (2011).
4
polarisation and misaligned voting results are.
In my benchmark model, parties in the legislature bargain only over policy and do not
discount the future. Here, if no party holds a majority of seats, the moderate party’s policy
will be implemented. While this enormous power of the moderate party may be debatable
in reality, the simplicity of bargaining underlines the novelty of the model’s voting stage.
Two clear predictions emerge from this benchmark model. First, when the moderate party
wins at least one seat, polarisation is mitigated: the policy of the left or right party can
only be implemented if a majority of voters in a majority of districts prefer it to the policy
of the moderate party. Second, if either the left or right party is a serious candidate in less
than half of the districts, there can be no misaligned voting. These results change somewhat
under different bargaining rules, but their flavour remains the same. When parties bargain
over perks of office as well as policy, the polarisation result is generally strengthened it is even more difficult to have extreme outcomes - while the misaligned voting result is
weakened - it can only be ruled out if a non-centrist party is serious in less than a quarter of
districts. When I add discounting to the benchmark model, the power of the moderate party
is reduced. Nonetheless, an extreme policy can still only be implemented if it is preferred to
a more moderate coalition policy. Here, misaligned voting cannot generally be eliminated in
all districts but may be ruled out in a large subset of districts.
Finally, when politicians are impatient, I show that if a country uses a random recognition
rule then coalition governments will be more difficult to construct than under a fixed order of
recognition, all else equal. The reason is that a fixed order rule gives a significant advantage
to the largest party and also makes it easier for voters to predict which government policy
will be implemented after the election. As the difference in policy between, say, a left
majority government and a coalition led by the left is quite small, voter preferences must
be very much skewed in favour of the left party in order for it to win a majority. With a
random recognition rule, however, risk averse voters will prefer the certainty of a non-centrist
single-party government to the lottery over policies which coalition bargaining would induce.
This paper contributes primarily to the theoretical literature on strategic voting in legislative elections. The bulk of this works has been on PR. Austen-Smith and Banks (1988)
find that, with a minimum share of votes required to enter the legislature, the moderate party
will receive just enough votes to ensure representation, with the remainder of its supporters
choosing to vote for either the left or right party. Baron and Diermeier (2001) show that,
with two dimensions of policy, either minimal-winning, surplus, or consensus governments
can form depending on the status quo. On plurality legislative elections Morelli (2004) and
Bandyopadhyay, Chatterjee, and Sjöström (2011) show that if parties can make pre-electoral
pacts, and candidate entry is endogenous, then voters will not need to act strategically. My
5
paper nonetheless focuses on strategic voting because in the main countries of interest, the
U.K. and Canada, there are generally no pre-electoral pacts and the three main parties
compete in almost every district, so strategic candidacy is not present.8
The paper proceeds as follows. In the next section I introduce the benchmark model and
define an equilibrium. In Section 3, I solve the model and show conditions which must hold
in equilibrium. Section 4 presents the main results on the level of polarisation and misaligned
voting in the benchmark model. Section 5 adds perks of office to the bargaining stage of the
model, while Section 6 shows how the benchmark results change when parties discount the
future. Finally, Section 7 discusses the assumptions of the model and concludes.
2
Model
Parties There are three parties; l, m, and r, contesting simultaneous elections in D districts, where D is an odd number. The winner of each of the D elections is decided by
plurality rule: whichever party receives the most votes in district d ∈ D is deemed elected
and takes a seat in the legislature. The outcome from all districts gives a distribution of seats
P
in the legislature, S ≡ (sl , sm , sr ), with party c ∈ {l, m, r} having sc seats and c sc = D.
Party c has a preferred platform ac in the unidimensional policy space X = [−1, 1] on which
it must compete in every district. A party cannot announce a different platform to gain
votes; voters know that a party will always implement its preferred platform if it gains a
legislative majority. Once all the seats in the legislature have been filled, the parties bargain
over the formation of government and implement a policy z. As such, a party cannot commit to implement its platform as the policy outcome z depends on bargaining. Party c has
the payoff Wc = bc − (z − ac )2 , linear in its share of government benefits bc , and negative
quadratic in the distance between its platform ac and the implemented policy z. A feasible
P
allocation of benefits is b = (bl , bm , br ) where each bc is non-negative and c bc ≤ B.
The benchmark model I use is that of Baron (1991), where B = 0 so that bargaining is
over policy alone.9 In Section 5, I consider a different bargaining game due to Austen-Smith
and Banks (1988) with B > 0. If a party has a majority of the seats in the legislature it
8
In the 2010 U.K. General Election, the three main parties contested 631 out of 650 districts (None of
them contest seats in Northern Ireland), while in the 2011 Canadian Federal Election the three major parties
contested 307 of the 308 seats.
9
A large literature has grown from legislative bargaining model of Baron and Ferejohn (1989), in which
legislators bargain over the division of a dollar. See Baron (1991), Banks and Duggan (2000), Baron and
Diermeier (2001), Jackson and Moselle (2002), Eraslan, Diermeier, and Merlo (2003), Kalandrakis (2004),
Banks and Duggan (2006), and many others. Morelli (1999) introduces a different approach to legislative
bargaining whereby potential coalition partners make demands to an endogenously chosen formateur. In
contrast with the Baron and Ferejohn (1989) setup, the formateur does not capture a disproportionate share
of the payoffs.
6
can form a unitary government and will implement its preferred policy. If no party wins
an outright majority we enter a stage of legislative bargaining. I consider two bargaining
protocols when B = 0: one in which the order of bargaining is random and one in which
it is fixed. Under the former rule, one party is randomly selected as formateur, where the
probability of each party being chosen is equal to its seat share in the legislature. The
formateur proposes a policy in [−1, 1], which is implemented if a majority of the legislature
support it; if not, a new formateur is selected, under the same random recognition rule,
and the process repeats itself until agreement is reached. Under the fixed order rule, the
party with the largest number of seats proposes a policy in [−1, 1], which is implemented if
a majority of the legislature support it; if not, the second largest party proposes a policy. If
this second policy does not gain majority support, the smallest party proposes a policy, and
if still there is no agreement, a new round of bargaining begins with the largest party again
first to move. I assume for now that parties are perfectly patient, δ = 1, but this is relaxed
in Section 6. A party’s strategy specifies which policy to propose if formateur, and which
policies to accept or reject otherwise.
Voters Individuals are purely policy-motivated with quadratic preferences on X. As such,
a voter does not care who wins his district per se, nor does he care which parties form
government; all that matters is the final policy, z, decided in the legislature. A voter’s type,
t ∈ T ⊂ X, is simply his position on the policy line; his utility is ut (z) = −(z − t)2 . I
assume T is sufficiently rich that for any tuple of distinct policies, (al , am , ar ), there is at
least one voter type who prefers one of the three over the other two. Furthermore, I assume
for simplicity that there is no type which is exactly indifferent between two platforms. Let
V ≡ {vl , vm , vr } be the set of feasible actions an individual can take, with vc indicating a
vote for party c. Voting is costless; thus, there will be no abstention.
Following Myerson (2000, 2002), the number of voters in each district d is not fully
known but rather is a random variable nd , which follows a Poisson distribution and has
−n k
mean n. The probability that there are exactly k voters in a district is P r[nd = k] = e k!n .
Appendix A summarises several properties of the Poisson model. The use of Poisson games
in large election models is now commonplace as it simplifies the calculation of probabilities
while still producing the same predictions as models with fixed but large populations.10
Each district has a distribution of types from which its voters are drawn, fd , which has
10
Krishna and Morgan (2011) use a Poisson model to show that in large elections, voluntary voting
dominates compulsory voting when voting is costly and voters have preferences over ideology and candidate
quality. Bouton and Castanheira (2012) use a Poisson model to show that when a divided majority need to
aggregate information as well as coordinate their voting behaviour, approval voting serves to bring about
the first-best outcome in a large election. Furthermore, Bouton (2013) uses a Poisson model to analyse the
properties of runoff elections.
7
full support over [−1, 1]. The probability of drawing a type t is fd (t). The actual population
of voters in d consists of nd independent draws from fd . A voter knows his own type, the
distribution from which he was drawn, and the distribution functions of the other districts,
f ≡ (f1 , . . . , fd , . . . , fD ), but he does not know the actual distribution of voters that is drawn
in any district.
A voter’s strategy is a mapping σ : T → ∆(V ) where σt,d (vc ) is the probability that a
type t voter in district d casts ballot vc . The usual constraints apply: σt,d (vc ) ≥ 0, ∀c and
P
σt,d (vc ) = 1, ∀t. In a Poisson game, all voters of the same type in the same district will
c
follow the same strategy (see Myerson (1998)). Given the various σt,d ’s, the expected vote
share of party c in the district is
τd (c) =
X
fd (t)σt,d (vc )
(1)
t∈T
which can also be interpreted as the probability of a randomly selected voter playing vc .
The expected distribution of party vote shares in d is τd ≡ (τd (l), τd (m), τd (r)). The realised
profile of votes is xd ≡ (xd (l), xd (m), xd (r)), but this is uncertain ex ante. As the population
of voters is made up of nd independent draws from fd , where E(nd ) = n, the expected number
of ballots for candidate c is E(xd (c)|σd ) = nτd (c). In the extremely unlikely event that
nobody votes, I assume that party m wins the seat.11 Let σ ≡ (σ1 , . . . , σd , . . . , σD ) denote
the profile of voter strategies across districts and let σ−d be that profile with σd omitted.
Let τ ≡ (τ1 , . . . , τd , . . . , τD ) denote the profile of expected party vote share distributions and
let τ−d be that profile with τd omitted. Thus, we have τ (σ, f ).
At this point, I could define an equilibrium of the game; however, it is more convenient to
define equilibrium in terms of pivotality and decisiveness, so I first introduce these additional
concepts below.
Pivotality, Decisiveness and Payoffs A single vote is pivotal if it makes or breaks a tie
for first place in the district. A district is decisive if the policy outcome z depends on which
candidate that district elects. When deciding on his strategy, a voter need only consider
cases in which his vote affects the policy outcome. Therefore, he will condition his vote
choice on being pivotal in his district and on the district being decisive. The ability to do
so is key, as if a voter cannot condition on some event where his vote matters then he does
not know how he should vote.
Let pivd (c, c0 ) denote when, in district d, a vote for party c0 is pivotal against c. This
occurs when xd (c) = xd (c0 ) ≥ xd (c00 ) – so that an extra vote for c0 means it wins the seat –
11
The probability of zero turnout in a district is e−n .
8
or when xd (c) = xd (c0 ) + 1 ≥ xd (c00 ) – so that an extra vote for c0 forces a tie. In the event
of a tie, a coin toss determines the winner.
Let λd denote an event in which district d is decisive in determining which policy z is
implemented; and let λid denote the i-th most likely decisive event for district d. Here the
decision of district d will lead to one of three final policy outcomes, thus, we can write each
i
decisive event as λid (z i ) or λid (zli , zm
, zri ) where each zci is the policy outcome of the legislative
bargaining stage when the decisive district elects party c. Note that these policies need not
correspond to the announced platforms of the parties - typically coalition bargaining will
i
lead to compromised policies. Two decisive events λid and λjd are distinct if (zli , zm
, zri ) 6=
j
, zrj ). Let Λ be the set of distinct decisive events; this set consists of I elements. As
(zlj , zm
we will see, the number and type of decisive events in the set Λ depends on the legislative
bargaining rule used.
It is useful for the following sections to classify decisive events into three categories. Let
i
, and zri are different points on the
λ(3) be a decisive event where all three policies zli , zm
policy line; let λ(2) be a case where two of the three policies are identical.12 Finally, let λ(20 )
be an event where there are three different policies but one of them is the preferred choice
of no voter.13 We can now turn to voter payoffs.
Let Gt,d (vc |nτ ) denote the expected gain for a voter of type t in district d of voting for
party c, given the strategies of all other players in the game – this includes players in his
own district as well as those in the other D − 1 districts. The expected gain of voting vl is
given by
Gt,d (vl |nτ ) =
I
X
P r[λid ]
i
i
i
i
P r[pivd (m, l)] ut (zl ) − ut (zm ) + P r[pivd (r, l)] ut (zl ) − ut (zr )
i=1
(2)
with the gain of voting vm and vr similarly defined. The probability of being pivotal between
two candidates, P r[pivd (c, c0 )], depends on the strategies and distribution of player types in
that district, summarised by τd , while the probability of district d being decisive depends
on the strategies and distributions of player types in the other D − 1 districts, τ−d . The
best response correspondence of a type t in district d to a strategy profile and distribution
of types given by τ is
BRt,d (nτ ) ≡ argmax
σt,d
X
σt,d (vc )Gt,d (vc |nτ )
(3)
vc ∈V
12
Obviously, λ(1) events cannot exist; if electing any of the three parties gives the same policy, it is not
a decisive event.
13
For this to be the case, the universally disliked policy must be a lottery over two or more policies.
9
Timing The sequence of play is as follows:
1. In each district, nature draws a population of nd voters from fd .
2. Voters observe platforms (al , am , ar ) and cast their vote for one of the three parties.
Whichever party wins a plurality in a district takes that seat in the legislature.
3. A government is formed according to a specified bargaining process and a policy outcome, z, is chosen.
Equilibrium Concept The equilibrium of this game consists of a voting equilibrium at
stage 2 and a bargaining equilibrium at stage 3. In a bargaining equilibrium, each party’s
strategy is a best response to the strategies of the other two parties. I restrict attention to
stationary bargaining equilibria, as is standard in such games.14
The solution concept for the voting game at stage 2 is strictly perfect equilibrium (Okada
(1981)).15 A strategy profile σ ∗ is a strictly perfect equilibrium if and only if ∃ > 0 such
∗
that ∀τ˜d ∈ ∆V : |τ˜d − τd (σ ∗ , f )| < then σt,d
∈ BRt,d (nτ̃ ) for all (t, d) ∈ T × D. That is,
the equilibrium must be robust to epsilon changes in the strategies of players. Bouton and
Gratton (2015) argue that restricting attention to such equilibria in multi-candidate Poisson
games is appropriate because it rules out unstable and undesirable equilibria identified by Fey
(1997). If, instead, Bayesian Nash equilibrium is used there may be knife-edge equilibria in
which voters expect two or more candidates to get exactly the same number of votes. Bouton
and Gratton (2015) also note that requiring strict perfection is equivalent to robustness to
heterogenous beliefs about the distribution of preferences, f . As I am interested in the
properties of large national elections, I analyse the limiting properties of such equilibria as
n → ∞.
3
Equilibrium
I solve for the equilibrium of the game by backward induction. The bargaining equilibrium
at stage 3 follows from Baron (1991). Of greater interest to us is the voting equilibrium at
stage 2. While there are multiple voting equilibria for any distribution of voter types, I show
that every equilibrium has only two candidates receiving votes in each district and I present
several properties which must hold in any equilibrium.
14
An equilibrium is stationary if it is a subgame perfect equilibrium and each party’s strategy is the same
at the beginning of each bargaining period, regardless of the history of play.
15
The original formulation of strictly perfect equilibrium was for games with a finite number of players;
Bouton and Gratton (2015) extend this to Poisson games.
10
Stage 3: Bargaining Equilibrium When no party has a majority of seats and δ = 1,
in any stationary bargaining equilibrium z = am is proposed and eventually passed with
probability one.16 This is regardless of whether the protocol is fixed order or random. To see
this, note that if any other policy is proposed, a majority of legislators will find it worthwhile
to wait until am is offered (which will occur when party m is eventually chosen as formateur).
The equilibrium policy outcome of the legislative bargaining stage is then


 al
z=
ar


am
if sl > D−1
2
if sr > D−1
2
otherwise
(4)
Every feasible seat distribution is mapped into a policy outcome, so, at stage 2, voters can
fully anticipate which policy will arise from a given seat distribution. The set of distinct
decisive events is given by
Λ = {λ(al , am , am ), λ(am , am , ar ), λ(al , am , ar )}
(5)
Stage 2: Voting Equilibrium A slight detour on how voters optimally cast their vote
is in order before describing the voting equilibrium. We know from the Magnitude Theorem
(Myerson (2000), see appendix) that as n → ∞ voters in a single district election need only
condition their choice on the most likely vote profile in which their vote is pivotal. The
following lemma extends this result to the case of multi-district elections considered here.
Lemma 1. As n → ∞ a voter need only condition his vote on the most likely event in which
his vote is both pivotal and decisive over two distinct policy outcomes.
Proof. See Appendix A.
An intuitive way of thinking of a voter’s decision process is the following. As in standalone plurality elections, a voter must consider the relative probabilities of his vote changing
the outcome in his district. However, he must also consider what happens in other districts.
Given τ−d , the voter will have an expectation about what the seat distribution will be before
his district votes, E(S−d ). If such an expectation means that d is not decisive, then the voter
will look to the most likely upset out of all the districts - where an expected winner in a
district instead loses. If d is now decisive, he can condition on this event; if not, he must
consider the next most likely upset. This processes of continues until the voter has worked
out the most likely chain of upsets which must occur in order to make his district decisive.
16
For a proof see Jackson and Moselle (2002).
11
Combining the probability of various upsets in the other districts with the probability of
being pivotal in district d, the voter can rank the probabilities of the various cases in which
his vote choice will change his utility. Only the relative probabilities of these events matter
for the voter. The lemma says that in a large election a voter need only condition his ballot
on the most likely of these events.
Now that we have analysed the decision problem of a single voter, we can see what
happens in an equilibrium. Voting games where players have three choices typically have
multiple equilibria; this game is no exception. However, I show that every voting equilibrium
involves only two candidates getting votes in each district.
Proposition 1. For any majoritarian legislative bargaining rule and any distribution of
voter preferences, f ≡ (f1 , . . . , fd , . . . , fD ), there are multiple equilibria; in every equilibrium
districts are duvergerian.
Proof. See Appendix A.
It is perhaps unsurprising that there are multiple equilibria and districts are always
duvergerian, especially given the findings of the extensive literature on single district plurality
elections. The logic as to why races are duvergerian is similar to the single district case; voters
condition their ballot choice on the most likely case in which they are pivotal, decisive and
not indifferent over outcomes. This greatly simplifies the decision process of voters and
means they need only consider the two frontrunners in their district.17 While we cannot pin
down which equilibrium will be played, the following properties will always hold.
1. In each district only two candidates receive votes; call these serious candidates.
2. If τd (c) > τd (c0 ) > 0, candidate c is the expected winner and his probability of winning
goes to one as n → ∞. Let dc denote such a district.
3. The expected seat distribution is E(S) = E(sl , sm , sr ) = (#dl , #dm , #dr ).
4. A district with c and c0 as serious candidates will condition on the most likely decisive
event λi ∈ Λ such that zci 6= zci0 .
The fourth property says: if a district’s most likely decisive event, λ1d , is of the type λ(3)
or λ(20 ), then voters must be conditioning on this event; if λ1d is of type λ(2), voters will be
conditioning on it only if they are not indifferent between the two serious candidates.
17
By restricting attention to strictly perfect equilibria we rule out knife edge cases where candidates are
expected to get exactly the same share of votes.
12
4
Analysis of Benchmark Model
This section presents the main results of the paper: the problems associated with standard
plurality rule election - polarisation and misaligned voting - can be mitigated in legislative
elections. Throughout the rest of the paper, several figures show the decision problem faced
by voters. These should make the arguments made in proofs easier to follow, hence, a slight
detour is needed to explain the graphical approach.
Recall, from Equation 5, that there are three distinct decisive events a district may
condition on:
Λ = {λ(al , am , am ), λ(am , am , ar ), λ(al , am , ar )}
The first two are λ(2) events while the final one is a λ(3) event. A λ(al , am , am ) event occurs
when a district is decisive in determining whether l wins a majority of seats and implements
z = al , or it falls just short, allowing a coalition to implement z = am . Here, voter types
m
≡ alm will vote vl while those of type t > alm will coordinate on either vm or
t < al +a
2
vr , as they are indifferent between the two policies offered. Similarly, in a λ(am , am , ar )
event, a district can secure party r a majority of seats or not; those with t < amr will vote
either vl or vm , while the rest will choose vr . Finally, when a district finds itself at a point
λ(al , am , ar ), it is conditioning on l and r winning half the seats each before d’s result is
, 0, D−1
). Electing either l or r would give them a majority, while
included: S−d = ( D−1
2
2
electing m would bring about a coalition. Therefore, depending on the result in d, any of
the three policies al , am or ar could be implemented.
These three distinct decisive events are represented in Figure 1. The simplex represents
the decision problem of voters in district d, holding fixed the strategies of those in the other
D − 1 districts.18 Each point corresponds to an expected distribution of D − 1 seats among
the three parties: the bottom left point corresponds to E(S−d ) = (D − 1, 0, 0); the bottom
right, E(S−d ) = (0, 0, D − 1); the apex is E(S−d ) = (0, D − 1, 0). For any given point, the
number of party m seats can be read directly off the y-axis, while the the number of party
r seats can be read by moving down the 45 degree line in a southwesterly direction to the
x-axis. The number of party l seats is simply the remainder.
Each district will condition on one of the distinct events λ ∈ Λ when voting, and this
must be consistent with the equilibrium properties given in the previous section. All dl
have E(S−dl ) = E(sl − 1, sm , sr ), all dm have E(S−dm ) = E(sl , sm − 1, sr ), and all dr have
E(S−dr ) = E(sl , sm , sr − 1). In Figure 1 this corresponds to the various E(S−d ) forming an
inverted triangle. An example is shown for the case of E(S) = (16, 2, 7).
18
While the simplex represents the case of D = 25, the same would hold for any odd D. To avoid the
case where two parties could share the seats equally, I ignore the case where D is even.
13
λ(al ,am ,am )
24
23
λ(am ,am ,ar )
22
21
λ(al ,am ,ar )
20
19
17
16
15
0
14
=
r
Pa
ty
r
13
l
ty
Pa
r
12
0
11
=
Number of Party m Seats
18
10
9
8
7
6
5
4
3
E(S-dr )
2
1
0
0
1
2
3
4
5
6
E(S-dl )
E(S-dm )
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Number of Party r Seats
Figure 1: Cases in which d is decisive when D = 25
4.1
Polarisation
Much attention in the U.S. has focused on how a system with two polarised parties has led
to policies which are far away from the median voter’s preferred point.19 An open question is
the degree to which policy outcomes reflect the preferences of voters in a legislative election
with three parties. Let t̃d be the expected position of the median voter in district d, and
label the districts so that t˜1 < . . . < t̃ D+1 < . . . < t̃D . Then, t̃ D+1 is the expected median
2
2
voter in the median district. While polarisation of outcomes can always occur in stand-alone
multi-candidate plurality elections, the following proposition shows that an extreme policy
can only be implemented in my setting if it is preferred to a moderate policy by the median
voter in the median district.
Proposition 2. For any distribution of voter preferences, where bargaining occurs over
policy and δ = 1, if E(sm ) > 0 then the expected outcome can be al only if the median voter
in the median district prefers policy al to policy am ; that is, if t̃ D+1 < alm . Similarly, a
2
necessary condition for E(z) = ar is t̃ D+1 > amr .
2
Proof. See Appendix A.
19
See McCarty, Poole, and Rosenthal (2008) as well as Polborn and Krasa (2015).
14
The focus on expected policies in the proposition is because of the random nature of the
model. It is always possible, though immensely unlikely, for the realised population of voters
to differ from the expected population sufficiently that E(z) 6= z. However, as n → ∞ this
probability goes to zero. The proposition stands in stark contrast to single district plurality
elections. In a stand-alone plurality election it can always be that l and r are the serious
candidates, so either al or ar will be implemented. A median voter will elect l as long as he
prefers al to ar . In a legislative election it takes much more to get non-moderate policies.
For al to come about it must be that (a) voters act as if their vote is pivotal in deciding
between an l majority government and a coalition, and (b) a majority of voters in a majority
of districts prefer that majority government policy, al , to the coalition policy.
This result gives a novel insight into multiparty legislative elections under plurality. In
the U.K., until recently, a vote for the Liberal Democrats (Lib Dems) has typically been considered a “wasted vote”.20 The popular belief was that the Lib Dems were not a legitimate
contender for government and so, even if they took a number of seats in parliament, they
would not influence policy. As a result, centrist voters instead voted for either the Conservatives or Labour. My model shows that electing a Lib Dem candidate is far from a waste.
Electing just one member of the Lib Dems to the legislature will be enough to moderate
extreme policies unless voter preferences favour one of the non-centrist parties very much.
In this benchmark case moderation leads to any coalition implementing Lib Dem policies in
full. In Section 5 and Section 6 we’ll see that a coalition doesn’t implement the exact policy
of the moderate party - nonetheless, coalition policies remain quite centrist and extreme
outcomes are still mitigated. This result suggests that concerns about the average voter not
being adequately represented in the U.K. or Canada are misplaced. If the Conservatives win
a majority in parliament it must be that a majority of voters in a majority of districts prefer
their policy to that of a moderate coalition. On the other hand, a moderate coalition can
come about for any distribution of voter preferences.21 Supporters of the Lib Dems in the
U.K. and the Liberals in Canada are therefore hugely advantaged by the current electoral
system; it is the supporters of the non-centrist parties who are disadvantaged.
4.2
Misaligned Voting
All voters are strategic: a voter chooses his ballot to maximise his expected utility; which
ballot this is depends on how the others vote. In any given situation, an individual may cast
20
See “What Future for the Liberal Democrats” by Lord Ashcroft, 2010.
For any f , there are always equilibria where z = am is the expected outcome. If support is strong
for party r then an equilibrium in which districts focus on a λ(al , am , am ) decisive event will give z = am .
Likewise, if l is popular then a focus on λ(am , am , ar ) will give z = am .
21
15
the same ballot he would if his vote unilaterally decided the district, or the strategies played
by the others in the district may mean his best response is to vote for a less desirable option.
Following Kawai and Watanabe (2013), I call the latter misaligned voting.
Definition. A voter casts a misaligned vote if, conditioning on the strategies of voters in
other districts, he would prefer a different candidate to win his district than the one he votes
for.
If a voter casts a misaligned vote, he is essentially giving up on his preferred candidate
due to the electoral mechanism. In a single plurality election there is only one district - so
there is no conditioning on other districts. With candidates l, m and r, whichever candidate
is least likely to be pivotal will be abandoned by his supporters, leading to a two-party race.
Therefore, either all types with t < alm , all types with t > amr , or those in the interval in
between will cast a misaligned vote. In contrast, Proposition 3 below shows that when δ = 1
there are many equilibria of the legislative election in which there is no misaligned voting in
any district.
Proposition 3. For any distribution of voter preferences, with bargaining over policy and
δ = 1, there always exist equilibria with no misaligned voting in any district. These occur
districts.
when l or r are serious candidates in fewer than D−1
2
Proof. See Appendix A.
The crux of the proposition is that when one of the non-centrist parties is a serious
candidate in less than half the districts, only one distinct decisive event exists. This event
is a choice over two policies; with only two policies on the table there is no strategic choice
to make - voters simply vote for their preferred option of the two. So, there can be no
misaligned voting. A voter with t > alm facing a λ(al , am , am ) decisive event is indifferent
between electing m or r; he will vote for whichever of the two the other voters are coordinating
on.
The proposition gives us a clear prediction on when there will and will not be misaligned
voting with three parties competing in a legislative election. It shows that the conventional
wisdom - no misaligned with two candidates, always misaligned with three - is wide of the
mark; whether there is misaligned voting or not depends on the strength of the non-centrist
parties. The proposition also has implications for the study of third-party entry into a twoparty system. Suppose, as is plausible, that a newly formed party cannot become focal in
many districts - maybe because they have limited resources, or because voters do not yet
consider them a serious alternative. Either way, an entering third-party is likely to be weaker
than the two established parties. Proposition 3 tells us that if a third party enters on the
16
flanks of the two established parties, then there will be no misaligned voting and no effect
on the policy outcome as long as this party is serious in less than half the districts. On the
other hand, if a third party enters at a policy point in between the two established parties,
this can shake up the political landscape. First of all, there will necessarily be misaligned
voting. Second of all, the policy outcome will now depend on which equilibrium voters focus
on - either al or am if t̃ D+1 < alr and am or ar if t̃ D+1 > alr . Success for the new party in
2
2
just one district can radically change the policy outcome. The implication is that parties
in a two-party system should be less concerned about the entry of fringe parties and more
concerned about potential centrist parties stealing the middle ground.
5
Legislative Bargaining over Policy and Perks
While the model of bargaining over policy in the previous section is tractable, it lacks
one of the key features of the government formation process: parties often bargain over perks
of office such as ministerial positions as well as over policy. Here, as parties can trade off
losses in the policy dimension for gains in the perks dimension, and vice versa, a larger set
of policy outcomes are possible. This section will show that, nonetheless, the results of the
benchmark model extend broadly to the case of bargaining over policy and perks.
The following legislative bargain model with B > 0 is due to Austen-Smith and Banks
(1988) (henchforth ASB).22 As usual, if a party wins an overall majority it will implement
its preferred policy and keep all of B. Otherwise, the parties enter into a stage of bargaining
over government formation. The party winning the most seats of the three begins the process
by offering a policy outcome y 1 ∈ X and a distribution of a fixed amount of transferable
private benefits across the parties, b1 = (b1l , b1m , b1r ) ∈ [0, B]3 . It is assumed that B is large
enough so that any possible governments can form, i.e. l can offer enough benefits to party r
so as to overcome their ideological differences. If the first proposal is rejected, the party with
the second largest number of seats gets to propose (y 2 , b2 ). If this is rejected, the smallest
party proposes (y 3 , b3 ). If no agreement has been reached after the third period, a caretaker
government implements (y 0 , b0 ), which gives zero utility to all parties.23
22
Other papers with bargaining over policy and perks include Diermeier and Merlo (2000) and Bandyopadhyay and Oak (2008).
23
This three-period protocol is a departure from the infinite horizon of the other bargaining rules I use.
Nonetheless, it is the standard ASB model and so is used for easy comparisons to the literature.
17
Seat Share
sl > (D − 1)/2
(D + 1)/2 > sl > sr > sm
(D + 1)/2 > sl > sm > sr
sm > sl , sr
(D + 1)/2 > sr > sm > sl
(D + 1)/2 > sr > sl > sm
sr > (D − 1)/2
3∆r < ∆l
al
alm
alr
am
am
ar
ar
2∆r < ∆l ≤ 3∆r
al
alm
alr
am
am
2am − alr
ar
∆r < ∆l ≤ 2∆r
al
alm
alr
am
am
amr
ar
∆l = ∆r
al
alm
am
am
am
amr
ar
∆l < ∆r ≤ 2∆l
al
alm
am
am
alr
amr
ar
2∆l < ∆r ≤ 3∆l
al
2am − alr
am
am
alr
amr
ar
3∆l < ∆r
al
al
am
am
alr
amr
ar
Table 1: Policy outcomes under ASB bargaining for any seat distribution and distance
between parties.
At its turn to make a proposal, party c solves
max B − bc0 − (y − ac )2
(6)
bc0 ,y
subject to bc0 − (y − ac0 )2 ≥ Wc0
where Wc0 is the continuation value of party c0 and Wc00 + (y − ac00 )2 > Wc0 + (y − ac0 )2 , so that
the formateur makes the offer to whichever party is cheaper. Solving the game by backward
induction, Austen-Smith and Banks (1988) show that a coalition government will always be
made up of the largest party and the smallest party. They solve for the equilibrium policy
outcome, for any possible distance between al , am and ar .
Table 1 shows the policy outcome for each seat distribution and distance between parties,
where ∆l ≡ am − al and ∆r ≡ ar − am . I assume if two parties have exactly the same number
of seats, a coin is tossed before the bargaining game begins to decide the order of play. So,
if sl = sr > sm , then with probability one-half, the game will play out as when sl > sr > sm
and otherwise as sr > sl > sm .
The set of possible policy outcomes depends on the number of seats in the legislature,
and on the distance between party policies. The simplex in Figure 2 shows what the policy
will be, for any seat distribution, when there are 25 districts and ∆l < ∆r ≤ 2∆l . Notice
that there are far more policy possibilities than in the case of B = 0. Figure 3 shows the
various decisive cases from the perspective of a single district; it is the analogue of Figure 1.
While there are many more decisive cases than when B = 0, they can be grouped into the
three categories defined previously: λ(2), λ(20 ) and λ(3) events.
The following proposition shows that, even when parties can bargain over perks as well
as policy, a non-centrist party will only win a majority if the median voter in the median
district prefers its policy to that of the centrist party.
Proposition 4. For any distribution of voter preferences, 3∆l > ∆r and E(sm ) > 1, the
expected outcome under ASB bargaining can be al only if t̃ D+1 < alm . Similarly, when
2
3∆r > ∆l and E(sm ) > 1, a necessary condition for E(z) = ar is t̃ D+1 > amr .
2
18
25
z =al
z =alm
z =E(alm ,am)
z =am
z =E(alm ,amr )
z =E(am ,alr )
z =alr
z =E(alr ,amr )
z =amr
z =ar
24
23
22
21
20
19
17
16
=0
15
l
rty
Pa
Pa
rty
r
14
13
0
12
=
Number of Party m Seats
18
11
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Number of Party r Seats
Figure 2: Policy outcomes under ASB bargaining, with D = 25 and ∆l < ∆r ≤ 2∆l
λ(2)
24
23
λ(2')
22
21
λ(3)
20
19
18
16
15
0
14
=
l
rty
Pa
Pa
rty
r
13
12
0
11
=
Number of Party m Seats
17
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Number of Party r Seats
Figure 3: Decisive events under ASB bargaining, with D = 25 and ∆l < ∆r ≤ 2∆l
19
Proof. See Appendix A.
Ruling out extreme outcomes here also requires that the non-centrist parties are not
exceptionally asymmetrically positioned. Given that restriction, we reaffirm the result of
Proposition 2, that moderate coalitions will be the norm in legislative elections unless the
population is heavily biased in favour of one of the non-centrist parties. Moreover, bargaining
over perks and policy can make it even more difficult for a non-centrist party to gain a
majority than in the benchmark case. This can be seen from Figure 2: starting from a point
, E(sm ) > 1 and D−1
< E(sr ) < D−1
, the most likely decisive event
E(S) where E(sl ) > D−1
2
4
2
for each district must be λ(al , alm , alm ). Therefore, a party l majority could only come about
lm
if t̃ D+1 < al +a
- an even stricter requirement than that of the benchmark case. This result
2
2
is noteworthy as in U.K. and Canadian elections party seat shares have tended to be in line
with this case: one of the non-centrist parties wins a majority, the other wins more than a
quarter of the seats, while the centrist party wins much less than a quarter.
On the other dimension of interest bargaining over policy and perks does not perform as
well; the restrictions needed to completely rule out misaligned voting are more severe than
in the benchmark model. However, as Proposition 6 will show, there are many equilibria in
which a large subset of districts have no misaligned voting.
Proposition 5. In a legislative election with ASB bargaining, there always exist equilibria
with no misaligned voting in any district.
1. When ∆l = ∆r , there is no misaligned voting if either party l or party r are serious in
fewer than D−1
districts.
4
2. When ∆l < ∆r , there is no misaligned voting if party r is serious in fewer than
districts.
D−1
4
3. When ∆l > ∆r , there is no misaligned voting if party l is serious in fewer than
districts.
D−1
4
Proof. See Appendix A.
The intuition is the same as in Proposition 3: when a non-centrist party is not serious
in enough districts, there is no hope of it influencing the outcome of legislative bargaining.
The threshold for relevance is lower than in the benchmark case. This is because the order
of parties matters for the policy outcome under ASB bargaining. From Figure 3 we see
that once it is possible for party r to win D−1
districts, two distinct decisive events exist:
4
λ(al , am , am ) and λ(al , am , alm ). No matter which of these two events a district focuses on,
20
and which two candidates are serious, some voters in the district will be casting misaligned
votes.
When party l or r have serious candidates in more than D−1
districts we cannot rule out
4
misaligned voting. This should not be surprising in a model with three differentiated parties
and multiple voter types on a policy line. We know that in single plurality elections there
will always be misaligned voting. What is surprising is that misaligned voting can ever be
ruled out in a district. The following proposition holds for all bargaining rules and gives
conditions for equilibria with no misaligned voting in a subset of districts.
Proposition 6. There will be no misaligned voting in a district if either
1. The most likely decisive event λ1d is a λ(20 ) event where candidates c and c0 are serious
and zc100 is preferred by no voter.
2. The most likely decisive event λ1d is a λ(2) event where candidates c and c0 are serious,
zc1 = zc100 , and all those voting vc must have ut (zci ) > ut (zci00 ) in the next most likely
decisive event λi ∈ Λ in which zci 6= zci00 .
Proof. See Appendix A.
The proposition is best understood by way of example. To see the first part, take a
, 2, D−3
) in Figure 3. Electing l will give sl > sr > sm
λ(2 ) event, for example, S−d = ( D−3
2
2
resulting in z = alm , electing r instead will give sr > sl > sm and bring about z = amr , while
electing m will lead to a tie for first place between l and r. A coin toss will decide which of
the two policies comes about, but ex ante voters’ expectation is E(alm , amr ). As voters have
concave utility functions, every voter strictly prefers either alm or amr to the lottery over the
pair. If this decisive event is the most likely (i.e. infinitely more likely than all others) and
the district focuses on a race between l and r, nobody in the district is casting a misaligned
vote.
To see the second part of the proposition, suppose the most likely decisive event is
, 3, D−5
). Here, electing l or m gives alm while electing r brings about a coin toss
S−d = ( D−3
2
2
and an expected policy E(alm , amr ). Suppose further that the second most likely decisive
event is S−d = ( D−5
, 3, D−3
), where electing m or r gives policy amr while electing l gives
2
2
E(alm , amr ). In the most likely event, all voters below a certain threshold will be indifferent
between electing l and electing m. However, in the second most likely decisive event, all
of these voters would prefer to elect l than m. Given that each decisive event is infinitely
more likely to occur than a less likely decisive event, these voters need only consider the
top two decisive events. Any voter who is indifferent between l and m in the most likely
decisive event strictly prefers l in the second most likely. So, if the district focuses on a race
0
21
between l and r there will be no misaligned voting. A special case of this second condition
is when λ1 is a λ(2) event and no λ2 event exists. This is what rules out misaligned voting
in Proposition 3 and Proposition 5.
Proposition 6 is quite useful, as it holds for any bargaining rule. It will allow me to
say that in the next section, even though we cannot get results such as Proposition 3 and
Proposition 5, we do not return to the single plurality election case of “always misaligned
voting”. Instead, there are again many equilibria in which a subset of districts have no
misaligned voting.
6
Impatient Parties
In this section, I examine how the results of the benchmark model change when δ < 1, so
that parties are no longer perfectly patient. It is likely that the discount rates of politicians
vary across countries depending on things such as constitutional constraints of bargaining,
the status quo, and the propensity of politicians to be reelected.24
In the benchmark model it didn’t matter whether the bargaining protocol was random
or had a fixed order; a coalition would always implement z = am . Once parties discount
the future, we get different policy outcomes depending on which of the two is used. Also,
once discounting is introduced into a bargaining model, one needs to decide whether players
receive payoffs at each stage of bargaining or only once an agreement is reached. In the former
case, the location of a status quo policy, Q, will be important for final policy outcomes. The
literature is far from united in the treatment of stage utilities in government formation.
With no stage payoffs, Jackson and Moselle (2002) show that there exists δ ∗ < 1 such that
if δ ≥ δ ∗ then the coalition policy will be within of am . However, their model doesn’t lend
itself easily to the current setup as mixing over proposals means precise policy outcomes
are not pinned down. Therefore, I follow Austen-Smith and Banks (2005) and Banks and
Duggan (2006) in having players receive stage payoffs. I assume the status quo is neither
too extreme, Q ∈ (−∆c , ∆c ) where ∆c = min{∆l , ∆r }, nor too central Q 6= am .25
In each period where no agreement is reached, the status quo policy remains and enters
party’s payoff functions. All parties discount the future at the same rate of δ ∈ (0, 1).
Therefore, if a proposal y is passed in period t, the payoff of party c is
Wc = −(1 − δ t−1 )(Q − ac )2 − δ t−1 (y − ac )2
24
(7)
After the 2010 Belgian elections, legislative bargaining lasted for a record-breaking 541 days, suggesting
high values of δ. Conversely, after the 2010 U.K. elections, a coalition government was formed within five
days.
25
If Q = am the result is the same as the benchmark case of δ = 1.
22
For ease of analysis I assume, without loss of generality, that am = 0.26 Banks and Duggan
(2000) show that all stationary equilibria are no-delay equilibria and are in pure strategies
when the policy space is one-dimensional and δ < 1.
6.1
Fixed Order Bargaining
The order of recognition is fixed and follows the ranking of parties’ seat shares. In
Appendix A, I derive the policy outcomes for any ordering of parties; these are presented in
Table 2 below along with policies for specific values of δ and Q. From the table we see that
the further party m moves down the ranking of seat shares, the further the coalition policy
moves away from am . Figure 4 shows the various policy outcomes for any seat distribution
in the legislature. Figure 5 shows the frequency of the three categories of decisive events.
Seat Share
sl > (D − 1)/2
(D + 1)/2 > sl > sr > sm
(D + 1)/2 > sl > sm > sr
sm > sl , sr
(D + 1)/2 > sr > sm > sl
(D + 1)/2 > sr > sl > sm
sr > (D − 1)/2
Policy
p al
− p(1 − δ 2 )Q2
− (1 − δ)Q2
p am = 0
2
p (1 − δ)Q
(1 − δ 2 )Q2
ar
δ = 0.95
|Q| = 0.5
< −0.5
−0.156
−0.112
0
0.112
0.156
> 0.5
δ = 0.95
δ = 0.9
|Q| = 0.25 |Q| = 0.5
< −0.25
< −0.5
−0.078
−0.218
−0.056
−0.158
0
0
0.056
0.158
0.078
0.218
> 0.25
> 0.5
Table 2: Policy outcomes with fixed order bargaining over policy and δ < 1.
The proposition below shows that when the bargaining protocol is fixed, parties discount
the future, and the status quo is not exactly am , it is even more difficult to have polarised
outcomes than is the case in the benchmark model.
Proposition 7. For any distribution of voter preferences, with a fixed order of bargaining
over policy, δ < 1 and E(sm ), E(sr ) > 1; the expected outcome can be al only if t̃ D+1 <
2
√
al − (1−δ)Q2
< alm . Similarly, when E(sl ), E(sm ) > 1; then a necessary condition for E(z) =
2
√
ar + (1−δ)Q2
> amr .
ar is t̃ D+1 >
2
2
Proof. See Appendix A.
Here, a majority government will only come about if the electorate is even more biased in
26
Taking any original positions (al , am 6= 0, ar ), we can always alter f so that the preferences of all voter
types are the same when (a0l , a0m = 0, a0r ).
23
25
z =E(-√(1-δ2)Q2,√(1-δ2)Q2 )
z =E(-√(1-δ2)Q2,-√(1-δ)Q2 )
z =E(√(1-δ)Q2,√(1-δ2)Q2 )
z =E(am ,√(1-δ)Q2)
z =E(-√(1-δ)Q2,am)
24
23
22
21
20
19
15
Pa
rty
r
14
=0
16
13
0
12
=
Number of Party m Seats
17
l
rty
Pa
z =√(1-δ)Q2
z =√(1-δ2)Q2
z =-√(1-δ2)Q2
z =-√(1-δ)Q2
z =am=0
z =ar
z =al
18
11
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Number of Party r Seats
Figure 4: Policy outcomes under fixed order bargaining, with D = 25 and δ < 1
λ(2)
24
23
λ(2')
22
21
λ(3)
20
19
17
16
15
0
14
=
l
rty
Pa
Pa
rty
r
13
12
0
11
=
Number of Party m Seats
18
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Number of Party r Seats
Figure 5: Decisive events under fixed order bargaining, with D = 25 and δ < 1
24
favour of policy al than in the benchmark case.27 The reason is that in the benchmark case
every coalition implements z = am , while with discounting and a fixed order protocol, the
largest party has an advantage in coalition negotiations and can use this to get an alternative
policy passed. Voters anticipate this power in coalition formation, so will only vote to bring
about an l majority if they prefer it to an l-led coalition.
For example, if al = −1, δ = 0.95 and |Q| = 0.25 then an l coalition would implement a
policy −0.078 or −0.056. Realising the advantage of party l in bargaining, voters will only
approve al if t̃ D+1 < −0.528, which is slightly to the left of the indifferent type alm in the
2
benchmark case. What the proposition also shows is that the further the status quo is from
am , the more likely we are to have coalition governments, all else equal. This is because a
more distant status quo gives the formateur even more bargaining power over the moderate
party. For example, if al = −1, δ = 0.95 and |Q| = 0.5 then an l majority implementing al
can only come about if t̃ D+1 < −0.556. Similarly, reducing the discount factor will strengthen
2
the bargaining hand of the formateur: setting δ = 0.9 means z = al requires t̃ D+1 < −0.579.
2
Figure 5 shows the frequency of the three types of decisive event for this bargaining rule.
While it is quite similar to Figure 3, the difference is that now there is no condition we
can impose to rule out misaligned voting. The decisive events of Figure 5 where one party
has no seats are λ(3) events. For any E(S) at least one of these λ(3) events can always
be conditioned on. If a district is conditioning on a λ(3) event there must be misaligned
voting in that district. Nonetheless, Proposition 6 still holds here: there are equilibria with
no misaligned voting in a subset of districts - an improvement on a single plurality election.
The following proposition summarises the state of misaligned voting under this bargaining
rule.
Proposition 8. In a legislative election with a fixed order of bargaining over policy and
δ < 1, there is always misaligned voting in some district. However, equilibria exist with no
misaligned voting in a subset of districts.
6.2
Random Recognition Bargaining
In each period one party is randomly selected as formateur, where the probability of
each party being chosen is equal to its seat share in the legislature, sDc . Party payoffs are
again given by Equation 7. As usual if a party has a majority of seats it will implement
its preferred policy. Following Banks and Duggan (2006), when no party has a majority I
27
It is worth mentioning that without the restriction to E(sr ) > 1 in the proposition, the threshold
becomes t̃ D+1 < alm as in the benchmark case. This is because some districts may then condition on
2
D−1
( D−1
2 , 2 , 0), and have l and m as serious candidates. In such a case l will win the district only if t̃ < alm .
25
look for an equilibrium of the form yl = am − Ω, ym = am , yr = am + Ω. Cho and Duggan
(2003) show that this stationary equilibrium is unique. As bargaining is only over policy,
any minimum winning coalition will include party m. When there was no discounting this
meant party m could always achieve z = am . Now, however, the presence of discounting
and Q 6= 0 allows parties l and r to offer policies further away from am , which party m will
nonetheless support. The moderate party will be indifferent between accepting and rejecting
an offer y when
δ(D − sm )
Wm (y) = −(Ω)2 = −(1 − δ)(Q)2 −
(Ω)2
(8)
D
which, when rearranged gives
s
(1 − δ)Q2
Ω=
(9)
m
1 − δ D−s
D
Table 3 shows the equilibrium offer each party will make when chosen as formateur. Notice
that the policies offered by l and r depend on the seat share of party m. The more seats
party m has, the closer these offers get to zero. The other thing to notice is that the policies
lie inside (−Q, Q).
Formateur
yl
ym
yr
−
q
q
Policy
Q2 > −|Q|
1−δ
m
1−δ D−s
D
am = 0
1−δ
2
m Q
1−δ D−s
D
< |Q|
Table 3: Policy proposals with random order bargaining over policy, δ < 1.
For a seat distribution such that no party has a majority, the expected policy outcome
from bargaining is
sl
E(z) = −
D
s
1−δ
Q2
m
1 − δ D−s
D
!
sm
sr
+
(0) +
D
D
s
1−δ
Q2
m
1 − δ D−s
D
!
(10)
An extra seat for any of the three parties will increase their respective probabilities of
being the formateur and so affect the expected policy outcome. Thus, every district always
conditions on a choice between three distinct (expected) policies. We also see that as sm
increases, the expected policy moves closer and closer to zero. This occurs for two reasons;
firstly because there is a higher probability of party m being the formateur, and secondly
because sm enters the policy offers of l and r; as sm increases the absolute value of these
policies shrink.
The proposition below shows that when the bargaining protocol is random, parties dis26
count the future, and the status quo is not exactly am , it is easier for a non-centrist party to
win a majority of seats and implement its preferred policy than is the case in the benchmark
model (though still more difficult than in a single district election).
Proposition 9. For any distribution of voter preferences, with a random order of bargaining
over policy, δ < 1 and E(sm ) > 0; the expected outcome can be al only if t̃ D+1 < zl∗ , where
2
alm < zl∗ < am , alr . Similarly, a necessary condition for E(z) = ar is t̃ D+1 > zr∗ , where
2
am , alr < zr∗ < amr .
Proof. See Appendix A.
The proposition implies that we should witness more majority governments than coalition
governments when the bargaining protocol is random. The reason is that with a random
recognition rule voters face uncertainty if they choose to elect a coalition. The implemented
policy will vary greatly depending on which party is randomly chosen as formateur. As
voters are risk averse, they find the certainty of policy provided by a majority government
appealing. The median voter in the median district need not prefer the policy of a noncentrist party to that of party m in order for the former to win a majority of seats.
For example, if al = −1, D = 101, δ = 0.95 and |Q| = 0.25 then S = (50, 1, 50) would
bring about a lottery over policies (−0.229, 0, 0.229). The uncertainty generated means a
voter with t < −0.474 would prefer to elect an l majority government, a type slightly to the
right of alm , the indifferent type in the benchmark case. If instead we had |Q| = 0.5 and kept
the other parameters unchanged voters would face a lottery over policies (−0.459, 0, 0.459);
a type t < −0.396 would now prefer an l majority to a coalition government.
The proposition stacks the deck against a coalition government. It gives us the right-most
, 0, D−3
). This almost-equal
type who might ever want an l majority, that is when S−d = ( D−1
2
2
split of seats between the l and r party ensures huge variance in the coalition policy, making
a single party government very appealing. It is worth considering when an l majority would
be preferred to a more balanced coalition. Keep the parameters as in the example above
and let a district condition on S−d = (50, 25, 25) with l and r being serious in the district. If
|Q| = 0.25 then a victory for r will bring about a lottery (−0.105, 0, 0.105). A voter in this
district will prefer al to this lottery if t < −0.508. If instead |Q| = 0.5, a victory for r will
bring about a lottery (−0.209, 0, 0.209). A voter in this district will prefer al to this lottery
if t < −0.509. In contrast to the proposition, these less extreme cases have t̃ D+1 < zl∗ < alm
2
meaning it is more difficult for al to come about than in the benchmark model.
Along with the previous propositions on polarisation, Proposition 9 shows that no matter which of the bargaining rules is used, there is less scope for polarisation in legislative
elections using plurality rule than there is in stand-alone plurality elections such as mayoral
27
24
23
λ(2')
22
21
λ(3)
20
19
17
16
15
0
14
=
l
rty
Pa
Pa
rty
r
13
12
0
11
=
Number of Party m Seats
18
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Number of Party r Seats
Figure 6: Decisive events under random recognition bargaining, with D = 25 and δ < 1
or presidential elections. This remains true even when the coalition formation process leads
to large uncertainty over policy outcomes.
Figure 6 shows the various decisive cases when the random recognition rule is used.
Almost all points are λ(3) events and, as we know, if a district is conditioning on such an
event it must have misaligned voting.28 There are, however, a selection of λ(20 ) events where
electing one of the three candidates is preferred by no voter. From Proposition 6 , if a district
is conditioning on such an event and the dominated candidate is not serious, there will be
no misaligned voting.
Proposition 10. For any distribution of voter preferences, in a legislative election with a
q
1−δ
D+1
:
random order of bargaining over policy, δ < 1, and where ζ ≡ D−3
1−δ D+1
2D
1. If |Q| > | aζl | or |Q| > | aζr | there exist equilibria with no misaligned voting in any district.
2. If |Q| < | aζl |, | aζr | there will be misaligned voting in at least one district.
Proof. See Appendix A.
28
The picture changes somewhat depending on the values of Q, D and δ: for certain values, decisive events
where sm is small may be λ(20 ) events or may even be events where all voters would like to elect m. However
this does not alter Proposition 10. Figure 6 shows the case of D = 25, δ = 0.99 and |Q| < 0.33.
28
If |Q| > | aζl | and each district has l and m as serious candidates then no voter type would
, D−1
, 1). Coalition
like to see party r win a seat. Electing r in a district would give S = ( D−1
2
2
bargaining would lead to a lottery over policies very close to policies al and am , so voters
would prefer the certainty of either a l or m majority government. This result contrasts
with the case of fixed order bargaining in Proposition 8. There the cases where a party is
expected to win no seats are what drives misalignment; Here, it is exactly these cases where
misaligned voting can be excluded.
For large D and δ we will have |Q| < | aζl |, | aζr |. However, we can still find equilibria with
no misaligned voting in as many as D − 1 districts. As I show in the proof, and as can be
the expected policy which comes about by electing the
seen from Figure 6, when sm = D−1
2
smallest party in the legislature is actually not preferred by any voter type. In this case,
if a district focuses on the two national frontrunners, there will be no misaligned voting.
We can even find cases where there will be misalignment in only a single district. If party
m has a majority of seats, party r has one seat, and l has the rest, then as long as all dm
and dl districts have m and l as serious candidates, only that single dr district will have
misaligned voting. This gives a fresh insight into the idea of “wasted votes”: if party l or r
is expected to be the smallest of the parties in the legislature, and party m is expected to
have a majority, then the least popular non-centrist party should optimally be abandoned
by voters. Any district which actually elects the weakest national party does so due to a
coordination failure; a majority there would instead prefer to elect one of the other two
parties. Notice, however, that for this to be the case, the moderate party must be expected
to win an overall majority. So, while the idea of a wasted vote does carry some weight, it
clearly does not apply to the case of the Liberal Democrats.
7
Discussion
In this paper, I introduced and analysed a model of three-party competition in legislative
elections under plurality rule. I showed that two properties of plurality rule - polarisation
and misaligned voting - are significantly reduced when the rule is used to elect a legislature.
The degree to which these phenomena are reduced depends on the institutional setup specifically, on how legislative bargaining occurs - but overall the results show that a plurality
rule electoral system reflects voter preferences much more than a single district model would
suggest.
In the benchmark model, parties are perfectly patient and bargain only over policy.
Two clear results emerged from this model. First, while an extreme policy can always
come about in standard plurality elections, in my setting a non-centrist policy needs broad
29
support in the electorate in order to be implemented; specifically, the median voter in the
median district must prefer the extreme policy to the moderate party’s policy. Second, while
standard plurality elections with three distinct choices always have misaligned voting, in my
benchmark model this is the case only if the non-centrist parties are serious candidates in
more than half the districts - otherwise there is no misaligned voting in any district.
The results of the benchmark model largely hold up under the other bargaining rules
considered: the non-centrist parties cannot win for any voter preferences (unlike in standard
plurality elections), and there are always equilibria in which there is no misaligned voting
(at least in a subset of districts). Moreover, if parties are impatient we gain an additional
insight: with a fixed order of formateur recognition we should see more coalitions while when
the order is random we should see more single-party governments, all else equal.
In a 2011 referendum UK voters were asked to choose between plurality rule and Instant
Run-off Voting (IRV) as a means of electing parliament.29,30 Supporters of IRV campaigned
by appealing to the undesirable properties of plurality rule; claiming plurality protected a two
party system of the non-centrist Conservative and Labour parties while the votes of millions
of centrist voters were wasted. This paper shows that applying what we know about single
district elections to legislative elections can lead to wrong conclusions. If we really want
to know whether plurality is outperformed by IRV (or PR for that matter) as an electoral
system then we need to analyse the full legislative election game. Such a comparison between
systems is beyond the scope of this paper and is left for future work.
In the remainder of this section I discuss the robustness of my modelling assumptions.
First, if utility functions are concave rather than specifically quadratic, the benchmark model
is unchanged. When bargaining also involves perks, the same is true as long as there are
enough perks to allow a coalition between the left and right parties to form.31 Second, if
parties bargain by making demands rather than offers, as in Morelli (1999), the results will
be the same as in the benchmark model.32 Third, if instead of a Poisson model I assumed a
fixed population size drawn from a multinomial distribution, the results of my model would
still go through.33
29
In the referendum, UK Voters decided to retain plurality rule.
IRV is used to elect the Australian Parliament. Under IRV voters rank the candidates in their district
on the ballot paper. If no candidate receives more than 50% of first preferences, those with the least first
preference votes are eliminated and their second preferences are distributed to the remaining candidates.
This continues until one candidate passes the 50% threshold and is elected.
31
If the perks are not large enough or parties don’t value perks enough, a coalition will always involve
the moderate party and we return to the simpler bargaining over policy case.
32
Whenever there is no clear majority, the head of state selects party m as the first mover, so the coalition
policy will be z = am .
33
Myerson (2000) shows that the magnitude of an event with a multinomial distribution is a simple
transformation of its magnitude with a Poisson distribution. This transformation preserves the ordering of
30
30
A key assumption is that voters only care about the policy implemented in the legislature.
If they also have preferences over who wins their local district, the results of the model no
longer hold: the probability of being pivotal locally would outweigh any possible utility
gain at the national level so that voters will only consider the local dimension. However, in
Westminster systems, a Member of Parliament has no power to implement policy at a local
level; he merely serves as an agent of his constituents: bringing up local issues in parliament,
helping constituents with housing authority claims, etc. So, if voters do have preferences
over their local winner, it should only be on a common-value, valence dimension. If this were
indeed the case, party policies would be irrelevant for how voters cast their ballots.
Finally, the assumption of perfect information is unrealistic in a real world election. The
asymptotic elements of the model mean voters can perfectly rank the probabilities of certain
events. In real life we are never that confident: polls may not be accurate, or more often,
polls may not exist at the district level. Myatt (2007) and Fisher and Myatt (2014) have
analysed single plurality elections with aggregate uncertainty over voters’ intentions. While
this paper abstracts from aggregate uncertainty for the sake of comparison with standard
models, including greater uncertainty in a multi-district model is an important path for
future research.
A
A.1
Appendix
Poisson Properties
The number of voters in a district is a Poisson random variable nd with mean n. The
−n k
probability of having exactly k voters is P r[nd = k] = e k!n . Poisson Voting games exhibit
some useful properties. By environmental equivalence, from the perspective of a player in
the game, the number of other players is also a Poisson random variable nd with mean n.
By the decomposition property, the number of voters of type t is Poisson distributed with
mean nfd (t), and is independent of the number of other player types. The probability of a
vote profile xd = (xd (l), xd (m), xd (r)) given voter strategies is
P r[xd |nτd ] =
Y
c∈{l,m,r}
e−nτd (c) (nτd (c))xd (c)
xd (c)!
(11)
events and so this means that the ordering of sets of pivotal and decisive events in my model would remain
unchanged, and therefore so would the equilibria.
31
Its associated magnitude is
X
log(P r[xd |nτd ])
xd (c)
µ(xd ) ≡ lim
= lim
τd (c)ψ
n→∞
n→∞
n
nτd (c)
(12)
c∈{l,m,r}
where ψ(θ) = θ(1 − log(θ)) − 1.
Magnitude Theorem Let an event Ad be a subset of all possible vote profiles in district
d. The magnitude theorem (Myerson (2000)) states that for a large population of size n, the
magnitude of an event, µ(Ad ), is:
log(P r[Ad ])
= lim max
µ(Ad ) ≡ lim
n→∞ xd ∈Ad
n→∞
n
X
τd (c)ψ
c∈{l,m,r}
xd (c)
nτd (c)
(13)
where ψ(θ) = θ(1 − log(θ)) − 1. That is, as n → ∞, the magnitude of an event Ad is simply
the magnitude of the most likely vote profile xd ∈ Ad . The magnitude µ(Ad ) ∈ [−1, 0]
represents the speed at which the probability of the event goes to zero as n → ∞; the more
negative its magnitude, the faster that event’s probability converges to zero.
Corollary to the Magnitude Theorem If two events Ad and A0d have µ(Ad ) > µ(A0d ),
then their probability ratio converges to zero as n → ∞.
P r[A0d ]
=0
n→∞ P r[Ad ]
µ(A0d ) < µ(Ad ) ⇒ lim
(14)
It is possible that two distinct events have the same magnitude. In this case, we must use
the offset theorem to compare their relative probabilities.
Offset Theorem Take two distinct events, Ad 6= A0d with the same magnitude, then
P r[Ad ]
=φ 0<φ<∞
n→∞ P r[A0 ]
d
µ(Ad ) = µ(A0d ) ⇒ lim
(15)
Suppose we have τd (c1 ) > τd (c2 ) > τd (c3 ), so that the subscript denotes a party’s expected ranking in terms of vote share. Maximising Equation 13 subject to the appropriate
32
constraints we get
µ(piv(i, j)) = µ(piv(j, i)) ∀i, j ∈ C
(16)
µ(c1 -win) = 0
p
µ(c2 -win) = µ(piv(c1 , c2 )) = 2 τd (c1 )τd (c2 ) − 1 + τd (c3 )
p
µ(c3 -win) = µ(piv(c2 , c3 )) = µ(piv(c1 , c3 )) = 3 3 τd (c1 )τd (c2 )τd (c3 ) − 1 if τd (c1 )τd (c3 ) < τd (c2 )2
p
µ(c3 -win) = µ(piv(c1 , c3 )) = 2 τd (c1 )τd (c3 ) − 1 + τd (c2 ) if τd (c1 )τd (c3 ) > τd (c2 )2
With a magnitude of zero, by the corollary, the probability of candidate c1 winning goes
to 1 as n gets large. Also, as the magnitude of a pivotal event between c1 and c2 is greater
than all other pivotal events, a pivotal event between c1 and c2 is infinitely more likely than
a pivotal event between any other pair as n gets large.
A.2
Proof of Lemma 1
To prove the lemma, I show that the magnitude theorem and its corollary extend to a
multi-district setting.
P
Step 1: µ(x) = µ(xd ).
d
Let x ≡ (x1 , . . . , xd , . . . , xD ) be the realised profile of votes across districts. The probability of a particular profile of votes is
P r[x|nτ ] =
Y
d∈D
c∈{l,m,r}
e−nτd (c) (nτd (c))xd (c)
xd (c)!
(17)
After some manipulation, taking the log of both sides, and taking the limit as n → ∞
we get the magnitude of this profile of votes
X X
log(P r[x|nτ ])
xd (c)
µ(x) ≡ lim
= lim
τd (c)ψ
n→∞
n→∞
n
nτd (c)
d∈D
(18)
c∈{l,m,r}
Notice that the magnitude of a particular profile of votes across districts is simply the sum
of the magnitudes in each district. This is because each district’s realised profile of votes
is independent of all other districts. So while µ(xd ) ∈ (−1, 0) in a single district, we have
µ(x) ∈ (−D, 0) when considering the profile of votes in all districts.
Step 2: If a particular profile of votes across districts, x, has a larger magnitude than
another, x0 , the former is infinitely more likely to occur as n → ∞.
33
The rate at which P r[x|nτ ] goes to zero is enµ(x) . Take another profile of votes across
districts x0 where µ(x0 ) < µ(x), then
0
P r[x0 |nτ ]
enµ(x )
0
= lim nµ(x) = lim en(µ(x )−µ(x)) = 0
n→∞ P r[x|nτ ]
n→∞ e
n→∞
lim
(19)
As their probability ratio converges to zero, x is infinitely more likely than x0 to occur as
n → ∞.
Step 3: Multi-District Magnitude Theorem. The magnitude theorem (Equation 13)
shows that the magnitude of an event occurring in a given district (such as a tie for first)
is simply to equal the magnitude of the most likely district vote profile in the set of district
vote profiles comprising that event. Here I extend this to the multi-district case. Let A =
(A1 , . . . , Ad , . . . , AD ) be a multi-district event, where each Ad is a particular district event.
P
xd (c)
, that is, x¯d is the most likely district vote
Let x¯d ∈ Ad = argmax
τd (c)ψ nτ
d (c)
xd
c∈{l,m,r}
profile in Ad given τd . Then, we have
µ(A) =
D
X
µ(Ad ) =
D
X
µ(x¯d ) = µ(x̄)
(20)
d=1
d=1
The first inequality follows from the independence of districts, the second equality follows
from the magnitude theorem and the independence of districts, the third equality follows
from Equation 18. Together they show that the single district magnitude theorem extends
to a multi-district setting.
Following from this, and using Equation 19, we have that the corollary to the magnitude
theorem also extends to the multi-district case. If µ(A0 ) < µ(A), then
0
P r[A0 |nτ ]
enµ(A )
0
lim
= lim nµ(A) = lim en(µ(x̄ )−µ(x̄)) = 0
n→∞ P r[A|nτ ]
n→∞ e
n→∞
(21)
Step 4: Voter’s Decision Problem
Let A(z, z 0 ) = {A(z, z 0 ), A0 (z, z 0 ), A00 (z, z 0 ), . . .} be the set of multi-district events
such that a single vote in district d is pivotal and decisive between policies z and
z0.
There are six different cases in which a vote may be pivotal and decisive
here: pivd (l, m)λ(z, z 0 , z 00 ), pivd (l, m)λ(z 0 , z, z 00 ), pivd (l, r)λ(z, z 00 , z 0 ), pivd (l, r)λ(z 0 , z 00 , z),
pivd (m, r)λ(z 00 , z, z 0 ), pivd (m, r)λ(z 00 , z 0 , z). Each case can occur from many different multidistrict events e.g A(z, z 0 ) may have xd0 (l) > xd0 (m) > xd0 (r) and xd00 (m) > xd00 (l) > xd00 (r)
while A0 (z, z 0 ) has the winners in d0 and d00 reversed.34 Using Equation 21 and Equation 20
34
From the perspective of a voter in district d the exact seats which are won in other districts don’t
matter, only their number.
34
we get
µ(A(z, z 0 )) = µ(Ā(z, z 0 )) = µ(x̄(z, z 0 ))
(22)
Where Ā(z, z 0 ) is the event in A(z, z 0 ) with largest magnitude, and x̄(z, z 0 ) is the vote profile
in Ā(z, z 0 ) with largest magnitude.
Let P ivΛd ≡ {A(z, z 0 ), A(z̃, z̃ 0 ), . . .} be the set of all distinct pivotal and decisive cases.
By Equation 22, comparing the elements in P ivΛd requires simply comparing the vote profiles
with largest magnitude in each A. From Equation 19 we know that if one vote profile has a
larger magnitude than another other it is infinitely more likely to occur as n goes to infinity.
Thus, if µ(A(z, z 0 )) > µ(A(z̃, z̃ 0 )) for all A(z̃, z̃ 0 ) ∈ P ivΛd then the former event is infinitely
more likely to occur. This being the case, the voter need only condition his vote choice on
the most likely vote profile x̄(z, z 0 ) ∈ P ivΛd .
A.3
Proof of Proposition 1
Step 1: All races are Duvergerian.
Note that if the decisive event with the largest magnitude λ1d is a λ(3) or λ(20 ) event then
it is immediate that the most likely pivotal and decisive event is pivd (c1 , c2 )λ1d . I now show
that this event is unique in a strictly perfect equilibrium.
Case 1: λ1d is a λ(3) or λ(20 ) event
• If in a given district τd (c1 ) = τd (c2 ) = τd (c3 ) = 31 then the most likely pivotal and
decisive event would not be unique, and we could support non-duvergerian results.
However, these equilibria are knife edge and do not survive when we adjust voter
strategies by .
• Similarly, any equilibrium in which τd (c1 ) = τd (c2 ) > τd (c3 ) > 0 or τd (c1 ) > τd (c2 ) =
τd (c3 ) > 0 is not robust to trembles and is thus ruled out by strict perfection.
• If P r[pivd (c1 , c2 )λ1d ] = P r[pivd (c1 , c2 )λ2d ], so that two distinct decisive events are equally
likely, then again a tremble on strategies in a district will ensure one decisive event is
more likely than the other.
Case 2: λ1d is a λ(2) event and zc11 = zc12 so voters are indifferent between the top two
candidates
• Suppose P r[pivd (c1 , c3 )λ1d ] = P r[pivd (c1 , c2 )λ2d ] so that some voters are conditioning on
the second most likely decisive event, while other voters in the district are conditioning
on the most likely decisive event and the imbalance is exactly offset by the fact that
35
the probability of a tie between c1 and c2 is higher than that between c1 and c3 . Here,
once again a tremble on players strategies in a district will break this equality, and the
race will become duvergerian.
• Can we have P r[pivd (c2 , c3 )λ1d ] = P r[pivd (c1 , c3 )λ1d ] > P r[pivd (c1 , c2 )λ2d ] ? We see from
Equation 16 that this is the only case in which two different pivotal events have the same
magnitude. That is when τ (c1 )τ (c3 ) < τ (c2 )2 so that µ(piv(c2 , c3 )) = µ(piv(c1 , c3 )) =
p
3 3 τ (c1 )τ (c2 )τ (c3 ) − 1. This occurs when the least popular candidate has much less
support than the other two. Trembles will do no good here for two reasons. Firstly, it
may not effect the behaviour of other districts if district d is a relatively safe seat i.e.
other districts are not conditioning on an upset occurring in d. Secondly, and more
importantly, the magnitudes of the pivotal events are not affected by such trembles
p
here; any tremble will leave 3 3 τ (c1 )τ (c2 )τ (c3 ) − 1 unchanged, so we have a candidate
for a strictly perfect equilibrium which is non-duvergerian. However, while these magnitudes are the same, we can use the offset theorem to see that their probabilities are
not. Specifically pivd (c1 , c3 ) 6= pivd (c2 , c3 ). We know from the offset theorem that
P r[pivd (c1 , c3 )]
=φ>0
P r[pivd (c2 , c3 )]
(23)
The magnitudes of these events are the same because the most likely event in which
they occur is when all three candidates get exactly the same number of votes. However,
the events consist of more than just this event. In an event pivd (c1 , c3 ) , candidate c2
must have the same or fewer votes than the others. Similarly, in an event pivd (c2 , c3 ) ,
candidate c1 must have the same or fewer votes than the others. By the decomposition
property, for any given number of votes c3 has, c1 is always more likely to have more
votes than c2 . Therefore, it must be that φ > 1. Returning to the decision of a voter
facing
µ(pivd (c1 , c3 )λ1d ) = µ(pivd (c2 , c3 )λ1d )
and given that λ1d is a λ(2) event, voters are indifferent between having c1 or c2 elected.
As I have just shown, P r[pivd (c1 , c3 )] > P r[pivd (c2 , c3 )], therefore all voters should
focus on this event, which will mean the previous second placed candidates loses all
her support to the leading candidate, thus ensuring a duvergerian equilibrium.
Therefore, all strictly perfect equilibria involve duvergerian races in every district.
Step 2: Multiple Equilibria always exist.
A simple example proves the existence of multiple pure strategy equilibria for any bargaining rule where a majority is needed to implement a policy z. For any f , suppose the
36
right party is never serious in any district. All races will be between l and m. We will
have E(S−dl ) = (k − 1, D − k, 0) and E(S−dm ) = (k, D − k − 1, 0) for some k ∈ (0, D). As
party r receives no votes, every district must be conditioning on the same decisive event
( D−1
, D−1
, 0). When conditioning on this λ, voters will face a choice between al , am , and
2
2
a third policy which would come about if r wins a seat. All the districts focusing on races
between l and m is an equilibrium as no individual would deviate and vote for r. Similarly, it is possible that all districts ignore the left party, and so an equilibrium will have
every district conditioning on (0, D−1
, D−1
), or that all districts ignore the moderate party
2
2
D−1
D−1
and all condition on ( 2 , 0, 2 ). These three equilibria always exist for any majoritarian
bargaining rule.
A.4
Lemma 2
A sufficient condition for µ(λid (z i )) > µ(λjd (z j )) is that the set of pivotal events with
largest magnitude required for λid (z i ) to occur, Kd (z i ), is a subset of those required for
λjd (z j ) to occur, Kd (z j ).
A.5
Proof of Lemma 2
From Equation 22 we have µ(λid (z i )) =
D−1
P
µ(x¯d0 ) ∈ (−D + 1, 0). That is the magnitude
d0 =1
of a particular decisive event, from the point of view of district d is simply the sum of
the magnitudes of the most likely vote profiles in each district d0 6= d which bring about
that decisive event. We can use µ(c1 − win) = 0 and µ(c2 − win) = µ(pivd0 (c1 , c2 )) from
Equation 16 and re-write
µ(λid (z i ))
=
k
X
µ(pivd0 (c1 , c2 )) ∈ (−k, 0)
d0 =1
Where there are k districts in which the expected winner loses. Let Kd (z i ) be the set of
these k pivotal events. Let another decisive event for district d be z j with magnitude
0
µ(λjd (z j ))
=
k
X
µ(pivd0 (c1 , c2 )) ∈ (−k 0 , 0)
d0 =1
Where there are k 0 districts in which the expected winner loses. Let Kd (z j ) be the set of
these k 0 pivotal events. Furthermore let Kd (z i ) ⊂ Kd (z j ). As magnitudes are negative
37
numbers, it must be that
k
X
d0 =1
µ(pivd0 (c1 , c2 )) >
k
X
0
µ(pivd0 (c1 , c2 )) +
d0 =1
k
X
µ(pivd0 (c1 , c2 ))
d0 =k+1
or
µ(λid (z i )) > µ(λjd (z j ))
(24)
Thus if Kd (z i ) ⊂ Kd (z j ) then µ(λid (z i )) > µ(λjd (z j )).
A.6
Corollary to Lemma 2
For a given E(S−d ) and Λ, we have λd (z) = λ1d if ∀λ(z 0 ) 6= λ(z) ∈ Λ we have Kd (z) ⊂
Kd (z 0 ).
A.7
Proof of Proposition 2
From Equation 4, for E(z) = al we must have an expected majority for the left party,
. By Proposition 1, each dl district must have either l and m or l and r as the
E(sl ) > D−1
2
serious candidates. Therefore, also by Proposition 1, if a dl district has λ1dl = λ(al , am , am ),
it must be conditioning on this decisive event. If a district d is conditioning on λ(al , am , am ),
the expected winner will be party l if the expected median voter prefers al to am , that is if
t˜d < alm . As party l is the expected winner in dl districts, any dl district conditioning on
then
λ(al , am , am ) must have t˜d < alm . If each dl has λ1dl = λ(al , am , am ) and E(sl ) > D−1
2
t̃ D+1 < alm .
2
All that remains is to show that when E(z) = al and E(sm ) > 0 each dl has λ1dl =
λ(al , am , am ). I show that case of E(sm ) = 1 as the case of E(sm ) > 1 is analogous.
Let Kdl (λ(am , am , ar )) be the sequence of pivotal events with largest magnitude resulting in λdl = λ(am , am , ar ). Let Kdl (λ(al , am , ar )) be the sequence of pivotal events with
+ k, 1, D−3
−
largest magnitude resulting in λdl = λ(al , am , ar ). Then given E(S−dl ) = ( D−1
2
2
k) it must be that Kdl (λ(al , am , am )) ⊂ Kdl (λ(am , am , ar )) and Kdl (λ(al , am , am )) ⊂
Kdl (λ(al , am , ar )). This can be seen graphically in Figure 1 as the pivotal events needed
to move from E(S−dl ) to λ(al , am , am ) are a subset of those needed to move from E(S−dl ) to
λ(am , am , ar ) or λ(al , am , ar ). By the Corollary to Lemma 2 this means λ1d = λ(al , am , am ).
It is analogous to show that a necessary condition for E(z) = ar is t̃ D+1 > amr .
2
38
A.8
Proof of Proposition 3
By Proposition 1, only two candidates will receive votes in each district. With D districts
there will be 2D serious candidates. If party r’s candidates are serious in less than D−1
2
districts, the decisive event in which an extra seat for party r gives them a majority can
never come about. Therefore, in any equilibrium where party r is serious in less than D−1
2
districts, the only decisive event voters can condition on is λ(al , am , am ). As this is the only
decisive event, in each district, voters with t < alm will vote vl while those with t > alm will
vote for whichever of m or r is a serious candidate. An analogous result holds when party l
districts.
is serious in less than D−1
2
A.9
Proof of Proposition 4
. As such
From Table 1 we see that when 3∆l > ∆r , z = al is only possible if sl > D−1
2
E(z) = al requires E(sl ) > D−1
. By Lemma 2, if E(sl ) > D−1
then λ1dl must be in the set of
2
2
decisive events Λsl = D−1 ⊂ Λ. In addition if E(sm ) > 1 then λ1dl must be in the set of decisive
2
events Λsm >1,sl = D−1 ⊂ Λsl = D−1 .
2
2
From Table 1 we see that when 3∆l > ∆r , any decisive event λi in the set Λsm >1,sl = D−1
2
i
, zri . From Proposition 1, each dl must have either l and r or else l and m
has zli = al 6= zm
as serious candidates. Therefore, with E(sl ) > D−1
, all dl districts must be conditioning on
2
decisive events in Λsm >1,sl = D−1 .
2
By the definition of a dl district, the median voter in dl must prefer zli = al to the some
i
or zri . The set of policies in Λsm >1,sl = D−1 are {al , alm , alr , 2am −alr , am }.
alternative: either zm
2
Suppose a district dl conditions on a race where the zli = al and the serious alternative is
am . It is immediate to see that t˜dl < alm . It is also immediate to see that if the serious
0
alternative was some other policy z 0 then t˜dl < al +z
. As am is the rightmost policy in the
2
set, a necessary condition for a dl district when E(sl ) > D−1
is t̃ < alm . Thus, it must be
2
that t̃ D+1 < alm in order for z = al to come about.
2
Following the same steps as above, it is analogous to show that when 3∆r > ∆l and
E(sm ) > 1, then E(z) = ar requires t̃ D+1 > amr .
2
A.10
Proof of Proposition 5
Case 1: ∆l = ∆r . Suppose party r is a serious candidate in less than D−1
districts; then
4
it can win at most that many seats. The possible election outcomes are either a party l
majority giving z = al , a party m majority giving z = am , or no majority but where party r
has the least seats. From Table 1 we see that when ∆l = ∆r , the policy outcome will be am
39
if no party has a majority and r is the smallest party. Conditional on r winning less than
D−1
districts, the only decisive event is therefore λ(al , am , am ), when l wins D−1
seats and
4
2
m is the second largest party. Given that only this distinct decisive event exists, all voters
must be conditioning on it. As electing m or r here brings about the same policy, voters are
indifferent between the two. In each district, those with t < alm will vote vl while those with
t > alm will coordinate on either vm or vr . With a choice over 2 policies in each district, no
voter will be casting a misaligned vote. The case of l being a serious candidate in less than
D−1
districts is analogous.
4
Case 2: ∆l < ∆r . The only difference from Case 1 we need to consider is when sr < D−1
4
D+1
and no party has a majority. From Table 1 we see that when ∆l < ∆r , then with 2 >
sl , sm > sr the policy outcome will be am . As shown above, when r is serious in less than
D−1
districts, the only distinct decisive case is λ(al , am , am ). All districts will condition on
4
this event and, as before, there is no misaligned voting.
Case 3: ∆l < ∆r . The possible election outcomes are either a party r majority giving
z = ar , a party m majority giving z = am , or no majority but where party l has the least
> sr , sm > sl the policy outcome
seats. From Table 1 we see that when ∆r < ∆l , if D+1
2
D−1
will be am . Therefore, when l is serious in less than 4 districts, the only distinct decisive
event is λ(am , am , ar ). All districts will condition on this event and there is no misaligned
voting.
A.11
Proof of Proposition 6
Case 1: Recall from Proposition 1 that if λ1d = λ(20 ) then voters in d must be conditioning
on it. Furthermore, by Lemma 1 this event is infinitely more likely to occur than any other
decisive event, and given that voters are not indifferent between any of the options, whichever
option a voter prefers in this case will also be his preferred over all possible decisive events.
By the definition of a λ(20 ) event, one of the 3 policy outcomes, zc100 , is dominated for each
voter by one of the two other policies, zc1 and zc10 . Therefore as long as zc100 is not serious,
there will be no misaligned voting - no voter would wish to change their vote if it could
unilaterally decide the district.
Case 2: Let the most likely decisive event λ1d be a λ(2) event where candidates c and
c0 are serious. If voters are conditioning on λ1d it must be that zc1 = zc100 or zc10 = zc100 ; Here,
I take it to be the former. Without loss of generality let zc1 < zc10 . Any voter type with
z 1 +z 1
t > c 2 c0 will vote vc0 , while any other type will vote vc . The former group cannot be
casting misaligned votes as they have ut (zc10 ) > ut (zc1 ), and the decisive event λ1d is infinitely
more likely than all others. Next, we need to consider whether any of the voters choosing
40
vc might be misaligned. All of these voters have ut (zc1 ) = ut (zc100 ) > ut (zc10 ), so that they
want to beat c0 but are indifferent between c and c00 in this most likely decisive event. If
one of these voters could unilaterally decide which candidate coordination takes place on,
he would decide by looking at the most likely pivotal event in which zc 6= zc00 , call this
event λi . If ut (zci ) > ut (zci00 ) then voter type t would prefer coordination to take place on
candidate c, while if ut (zci ) < ut (zci00 ) she’d want coordination on c00 . Therefore, there is
no misaligned voting in the district if there exists no type such that ut (zci ) < ut (zci00 ) and
ut (zc1 ) = ut (zc100 ) > ut (zc10 ) when c and c0 are the serious candidates.
A.12
Bargaining Equilibrium for Fixed Order Protocol and δ < 1
As equilibria are stationary we need only consider two orderings: l > r > m > l > r > . . .
and r > l > m > r > l > . . .. I will derive the equilibrium offers for the case of l > r > m,
the other is almost identical. I solve the game by backward induction. At stage 3, party m
will make an offer ym which maximises its payoff subject to the proposal being accepted by
either party l or r.
At stage 2, party r will either make an offer yr (m) to attract party m, or an offer yr (l)
to attract party l. For these proposals to be accepted by m and l respectively requires
2
−yr (m)2 ≥ −(1 − δ)Q2 − δym
−(al − yr (l))2 ≥ −(1 − δ)(al − Q)2 − δ(al − ym )2
p
2 .
If yr (m) is chosen then the first inequality will bind and we have yr (m) = (1 − δ)Q2 + δym
We can now compare the payoff of party l when yr (m) and yr (l) are implemented.
−(al −
p
p
2 )2 = −a2 − (1 − δ)Q2 − δy 2 + 2a
2 )
(1 − δ)Q2 + δym
(1 − δ)Q2 + δym
l
l
m
2
−(al − yr (l))2 = −a2l − (1 − δ)Q2 − δym
+ (1 − δ)2al Q + δ2al ym
Party l prefers policy yr (l) when
p
2 )
(1 − δ)Q2 + δym
p
2 )
2al ((1 − δ)Q + δym ) > 2al (1 − δ)Q2 + δym
p
2 )
(1 − δ)Q + δym < (1 − δ)Q2 + δym
(1 − δ)2al Q + δ2al ym > 2al
41
the final inequality always holds. As party l gets a higher payoff from yr (l) than yr (m), the
former must be closer to al on the policy line, and therefore further away from ar . Clearly
p
2 .
then, party r maximises its utility by choosing yr = (1 − δ)Q2 + δym
At stage 1, party l will either make an offer yl (m) to attract party m, or an offer yl (r) to
attract party r. For these proposals to be accepted by m and r respectively requires
p
2 )2
(1 − δ)Q2 + δym
p
2 )2
−(ar − yl (r))2 ≥ −(1 − δ)(ar − Q)2 − δ(ar − (1 − δ)Q2 + δym
−yl (m)2 ≥ −(1 − δ)Q2 − δ(−
If yl (m) is chosen then the first inequality will bind and we have yl (m) =
p
2 . We can now compare the payoff of party r when y (m) and y (r)
− (1 − δ 2 )Q2 + δ 2 ym
l
l
are implemented.
p
2 )
(1 − δ 2 )Q2 + δ 2 ym
p
2
2
−(ar − yl (r))2 = −(1 − δ)(a2r + Q2 − 2ar Q) − δa2r − δ(1 − δ)Q2 − δ 2 ym
+ δ2ar (1 − δ)Q2 + δym
−(ar +
p
2 )2 = −a2 − (1 − δ 2 )Q2 − δ 2 y 2 − 2a
(1 − δ 2 )Q2 + δ 2 ym
r
r
m
Party r prefers policy yl (r) when
p
p
2 > −2a
2 )
(1 − δ)Q2 + δym
(1 − δ 2 )Q2 + δ 2 ym
r
p
p
2 > −
2 )
(1 − δ)Q + δ (1 − δ)Q2 + δym
(1 − δ 2 )Q2 + δ 2 ym
(1 − δ)2ar Q + δ2ar
the final inequality always holds. As party r gets a higher payoff from yl (r) than yl (m), the
former must be closer to ar on the policy line, and therefore further away from al . Clearly
p
2 .
then, party l maximises its utility by choosing yl = − (1 − δ 2 )Q2 + δ 2 ym
Now, we can return to stage 3 to show that ym = 0. By stationarity, if ym is rejected at
p
2 will be proposed and accepted. Parties
stage 3, then in stage 4 yl = − (1 − δ 2 )Q2 + δ 2 ym
l and r will accept proposal ym if
p
2 )2
(1 − δ 2 )Q2 + δ 2 ym
p
2 )2
−(ar − ym )2 ≥ −(1 − δ)(ar − Q)2 − δ(ar + (1 − δ 2 )Q2 + δ 2 ym
−(al − ym )2 ≥ −(1 − δ)(al − Q)2 − δ(al +
Party m’s payoff is maximised when ym = 0 (because am = 0), so we want to check whether
42
this is an implementable proposal. Letting ym = 0 and rearranging, the two inequalities
above become
p
(1 − δ 2 )Q − (1 − δ)Q]
p
0 ≤ (1 − δ 3 )Q2 + 2ar [δ (1 − δ 2 )Q − (1 − δ)Q]
0 ≤ (1 − δ 3 )Q2 + 2al [δ
The term in square brackets may be positive or negative. If it is positive then, party r will
accept ym = 0, if the term is negative then party l will accept ym = 0.35
Given ym = 0, we can now characterise the accepted policy proposals (and therefore
policy outcomes) for the fixed order protocol when l > r > m > l > r > . . ..
p
yl = − (1 − δ 2 )Q2
p
yr =
(1 − δ)Q2
ym = 0
Instead when r > l > m > r > l > . . ., the same process gives:
yl
p
(1 − δ 2 )Q2
p
= − (1 − δ)Q2
yr =
ym = 0
A.13
Proof of Proposition 7
For z = al to be the expected outcome it must be that E(sl ) > D−1
. Given the restriction
2
that E(sm ), E(sr ) > 1, the set of distinct decisive events which dl districts can be conditioning
on is reduced to
p
p
Λsl = D−1 ,sm >1,sr >1 = {λ(al , − (1 − δ 2 )Q2 , − (1 − δ 2 )Q2 ),
2
p
p
λ(al , − (1 − δ 2 )Q2 , − (1 − δ)Q2 ),
p
p
λ(al , − (1 − δ)Q2 , − (1 − δ)Q2 )}
35
Whenever δ > 0.543689 then the term is positive. Given that we mostly care about values of δ close to
one, we can say that it is generally party r who accepts m’s offer.
43
p
Any race between
a
and
−
(1 − δ)Q2 , where the former is the expected winner, must
l
√
p
2
a − (1−δ)Q
.
Any
race
between
a
and
−
(1 − δ 2 )Q2 , where the former is the
have t̃ < l
l
2
√
a − (1−δ 2 )Q2
expected winner, must have t̃ < l
, a stricter condition. Therefore in order to
2
for a party l to √
win a majority in expectation when E(sm ), E(sr ) > 1 it must
√ be2 at least
p
2
a − (1−δ)Q
a − (1−δ)Q
. Notice that since − (1 − δ)Q2 < am , then l
< alm .
that t̃ D+1 < l
2
2
2
Similarly, for party
√ r to win a majority in expectation when E(sm ), E(sl ) > 1 it must be
that t̃ D+1 >
2
A.14
ar +
(1−δ)Q2
2
> amr .
Proof of Proposition 9
Under this bargaining rule, we can only have E(z) = al if E(sl ) > D−1
. If this is the case,
2
any dl district must be conditioning on a decisive event in Λsl = D−1 . Each of these decisive
2
events are distinct: by increasing a party’s seat share by one, it alters the expected policy
outcome E(z). In order to find the weakest condition for E(z) = al to occur, I proceed in the
following steps. Steps 1 and 2 show that electing party l is always the worst option for a player
, 1, D−1
) gives the lowest utility for any t < 0 type
with t < 0. Step 3 shows that S = ( D−1
2
2
conditional on sm > 0. Step 4 identifies the type who is indifferent between S = ( D−1
, 1, D−1
)
2
2
q
q
1−δ
1−δ
2
and a party l majority. Let Ω(r) ≡ 1−δ D−sm Q2 and Ω(m) ≡
D−(sm +1) Q . Note that
D
1−δ
D
Ω(m) < Ω(r).
i
)) or ut (al ) > ut (E(zri )), it must be that t < 0. I show the case
Step1: If ut (al ) > ut (E(zm
i
)) as the other is analogous. Comparing the expected utility of a voter
of ut (al ) > ut (E(zm
voting for l or m where sl = D−1
, sm > 0 and sr > 0 we have:
2
ut (l) = −(al − t)2
D−1
sm + 1 2 D − 1 − 2sm
ut (m) = −
(−Ω(m) − t)2 −
(t) −
(Ω(m) − t)2
2D
D
2D
Some algebra shows that a player has ut (l) > ut (m) if
t<
Da2l − (D − sm )Ω(m)2
2Dal − (1 − sm )2Ω(m)
The RHS must always be negative. It is easy to see that this would also be the case comparing
ut (l) to ut (r). Thus any player type who prefers al to a lottery over coalition policies must
have t < 0.
i
Step 2: At any decisive event λi ∈ Λsl = D−1 ,sr > 0 every t ≤ 0 prefers E(zm
) to E(zri ).
2
Comparing the expected utility of a voter voting for m or r for any case where sl = D−1
,
2
44
sm > 0 and sr > 0 we have:
D−1
sm + 1 2 D − 1 − 2sm
(−Ω(m) − t)2 −
(t) −
(Ω(m) − t)2
2D
D
2D
sm 2 D + 1 − 2sm
D−1
(−Ω(r) − t)2 −
(t) −
(Ω(r) − t)2
ut (r) = −
2D
D
2D
ut (m) = −
Some algebra shows that a player has ut (m) > ut (r) if
(D − sm )(Ω(r)2 − Ω(m)2 ) + Ω(r)2
t<
2(sm − 1)(Ω(r) − Ω(m)) + 2Ω(m)
i
) to
The RHS must always be positive. This means that any type t ≤ 0 always prefers E(zm
i
E(zr ).
Step 3: S = ( D−1
, 1, D−1
) gives the lowest utility for any t < 0 type conditional on
2
2
sm > 0. Using the result from Step 2 and noting that E(z|S−dr = ( D−1
, k, D−1−2k
)) =
2
2
D−1
D+1−2k
E(z|S−dm = ( 2 , k − 1, 2 )) we can see that for a t ≤ 0 the least preferred expected
policy is E(z|S = ( D−1
, 1, D−1
)).
2
2
, 1, D−1
)). To find the
Step 4: Identifying the type who prefers al to E(z|S = ( D−1
2
2
D−1
rightmost type who prefers al to a coalition we thus examine S = ( 2 , 1, D−3
). At this
2
point, electing party l gives them a majority and brings about z = al , while electing party
r leads to a coalition with an ex ante expected policy of
D−1 D−1
D−1
, 1,
) =−
E z|S = (
2
2
2D
s
1−δ
Q2
1 − δ D−1
D
!
2
D−1
+
(0) +
2D
2D
s
1−δ
Q2
1 − δ D−1
D
A voter will prefer the former if
D−1
−(al − t)2 > −
2D
s
−
!2
D−1
1−δ
2
Q2 − t −
(−t)2 −
D−1
2D
2D
1−δ D
s
!2
1−δ
Q2 − t
1 − δ D−1
D
rearranging this we get that a voter prefers al if
al D − 1
t< −
2
2Dal
1−δ
Q2
D−1
1 − δ( D )
!
As al < 0, the right hand side is greater than a2l , which is the cutoff point in the benchmark
case (recalling that alm = a2l when am = 0). The cutoff for a party l majority is thus given
45
!
by
t̃ D+1
2
al D − 1
< zl∗ ≡ −
2
2Dal
1−δ
Q2
1 − δ( D−1
)
D
!
> alm
(25)
As zl∗ > alm , the above result must also hold when sr = 0 as well as sr > 0.
Next we show that zl∗ < am , alr . First, I show zl∗ < am . Some manipulation shows
D−1
2Dal
Let P ≡
D−1
−δ D−1
D
D
D−1
1
−δ D−1
+D
D
D
1−δ
Q2
D−1
1 − δ( D )
!
=
Q2
2al
D−1
D
D−1
−
D
− δ D−1
D
D−1
δ D +
1
D
>
Q2
2al
(26)
where the inequality in Equation 26 comes from P < 1. In order to
2
(Q)
show zl∗ < am we need to show a2l − P 2a
< 0. Given P < 1 and |Q| ≤ |al |, this immediately
l
follows.
Second, I show zl∗ < alr . If ∆l < ∆r then necessarily alr > am , so the proof above applies.
If ∆l > ∆r we have zl∗ < alr if
al ar
al P (Q)2
−
< +
2
2al
2
2
2
ar
P (Q)
<
2|al |
2
As Q2 ≤ a2r and |al | > |ar |, the inequality above must hold.
Therefore, we havethat a majority
for party l , which brings about al can only occur if
a
D−1
1−δ
t̃ D+1 < zl∗ ≡ 2l − 2Da
Q2 where zl∗ < am , alr .
1−δ( D−1 )
l
2
ar
2
D
Analogously, a majority for party r , which brings about ar can only occur if t̃ D+1 > zr∗ ≡
2
D−1
1−δ
2
∗
+ 2Dar 1−δ( D−1 ) Q where zr > am , alr .
A.15
D
Proof of Proposition 10
Case 1: If −|Q| < aζl or aζr < |Q| there exist equilibria with no misaligned voting
in any district. When E(sl ) > D−1
and E(sr ) = 0, voters will be conditioning on the point
2
( D−1
, D−1
, 0). Electing l or m brings about z = al or z = am = 0 respectively, each of which
2
2
are the preferred policies of t = −1 and t = 0. Electing r leads to an expected coalition
policy given by
D−1
ut (r) = −
2D
s
−
!2
1−δ
D−1 2
2
Q2 − t −
(t) −
D+1
2D
2D
1 − δ 2D
46
s
!2
1−δ
Q2 − t
(27)
1 − δ D+1
2D
Some algebra shows that there exists a type for which Equation 27 is greater than both ut (l)
and ut (m) if
al
(28)
− |Q| >
ζ
q
1−δ
where ζ ≡ D+1
. Otherwise, if −|Q| < aζl , then ut (r) is dominated for all voter
D−3
1−δ D+1
2D
types, making ( D−1
, D−1
, 0) a λ(20 ) event. If each district has l and m as serious candidates
2
2
then there will be no misaligned voting. It is analogous to show that if aζr < |Q| < ar and
all districts have m and r as serious candidates, there is no misaligned voting.
Case 2: If |Q| > | aζl |, | aζr | there will be misaligned voting in at least one district.
Case 2A: When sl > D−1
or sr > D−1
there will always be misaligned voting in some
2
2
districts. I examine the case where l is expected to win a majority; the other case is identical.
, E(sr ) > 0 and E(sm ) > 0 there will be misaligned voting. Voters
1) When E(sl ) > D−1
2
; the expected utility of electing the
must be conditioning on a decisive event where sl = D−1
2
three different candidates is
ut (l) = −(al − t)2
D−1
ut (m) = −
2D
ut (r) = −
D−1
2D
!2
!
s
1
−
δ
s
+
1
D
−
1
−
2s
m
m
−
Q2 − t −
(t)2 −
Q2 − t
D−(sm +1)
D−(sm +1)
D
2D
1−δ
1−δ
D
D
!2
!2
s
s
D
+
1
−
2s
s
1−δ
1
−
δ
m
m
2
−
(t) −
Q2 − t −
Q2 − t
D−sm
m
D
2D
1 − δ D−s
1
−
δ
D
D
s
1−δ
It suffices to consider the most extreme types t = −1, t = 0 and t = 1. Subbing these values
in we see that a type t = −1 will always want to elect l and a type t = 0 will always want to
elect m. For any case E(sl ) > D−1
, E(sr ) > 0, E(sm ) > 0, there must therefore be misaligned
2
voting as in the districts where r is expected to win, the other voters coordinate on either l
or m. The supporters of that candidate which is not serious must be casting misaligned votes.
2) When E(sl ) > D−1
and E(sm ) = 0 there will be misaligned voting in every district.
2
Voters will be conditioning on the point ( D−1
, 0, D−1
) where electing l or r brings about
2
2
z = al or z = ar respectively, each of which are the preferred policies of t = −1 and t = 1.
Electing m leads to an expected coalition policy given by
D−1
ut (m) = −
2D
s
−
1−δ
Q2 − t
1 − δ D−1
D
!2
2
D−1
−
(t)2 −
2D
2D
s
!2
1−δ
Q2 − t
1 − δ D−1
D
Subbing in for t = 0, it is clear that u0 (m) > u0 (l), u0 (r). As only 2 candidates receive votes,
47
one of these player types must be casting a misaligned vote.
and E(sr ) = 0 there will be misaligned voting when −ζ|Q| > al .
3)When E(sl ) > D−1
2
This follows directly from Case 1.
Case 2B: When no party has an expected majority there will be misaligned voting in some
districts. For any expected seat distribution where no party has a majority, the expected
utility of electing the three different candidates is
!2
1−δ
sm 2 sr
−
Q2 − t −
(t) −
D−sm
D
D
1−δ D
!2
s
1−δ
sm + 1 2
−
Q2 − t −
(t) −
D−(sm +1)
D
1−δ
D
!2
s
1−δ
sm 2 sr + 1
−
Q2 − t −
(t) −
D−sm
D
D
1−δ D
sl + 1
ut (l) = −
D
ut (m) = −
sl
D
ut (r) = −
sl
D
s
!2
1−δ
Q2 − t
m
1 − δ D−s
D
!2
s
1−δ
sr
Q2 − t
D−(sm +1)
D
1−δ
D
!2
s
1−δ
Q2 − t
m
1 − δ D−s
D
s
Where I abuse notation slightly to let sc to be the expected number of seats of party
c before district d votes, so that sl + sm + sr = D − 1. By subbing in t = 0, we see
that this type will always want m elected. For there to be no misaligned voting it must
therefore be the case that m is a serious candidate in every district. Suppose this is the
case so that in a dr district r and m are serious candidates and in a dl district l and
m are the serious candidates. In a dr district a type t = 1 must have u1 (r) > u1 (m).
Note that a dr district conditions on r having one less seat and l having one more seat
than a dl district conditions on (Graphically, E(S−dr ) is one point to the left of E(S−dl )).
Using the equations above one can show that if u1 (r|(sl , sm , sr )) > u1 (m|(sl , sm , sr )) then
u1 (r|(sl − 1, sm , sr + 1)) > u1 (m|(sl − 1, sm , sr + 1)). That is, in a given equilibrium, if a
type t = 1 in a dr district prefers r to m, then a type t = 1 in a dl district also prefers r to
m. As this type also prefers r to l and the focal candidates in the dl district are l and m,
he must be casting a misaligned vote.
Case 2C: When sm > D−1
there will be misaligned voting in at least one district.
2
D−1
If sm > 2 , all districts must be conditioning on decisive events where sm = D−1
. In
2
48
such cases the expected utility of a type t voter in voting for each of the parties is
sl + 1
ut (l) = −
D
s
−
1−δ
Q2 − t
1 − δ D+1
2D
!2
D − 1 2 D − 1 − 2sl
−
(t) −
2D
2D
s
!2
1−δ
Q2 − t
1 − δ D+1
2D
ut (m) = − (t)2
sl
ut (r) = −
D
s
−
!2
1−δ
D − 1 2 D + 1 − 2sl
(t) −
Q2 − t −
D+1
2D
2D
1 − δ 2D
s
1−δ
Q2 − t
1 − δ D−1
2D
!2
Note that for t = 0 we have ut (m) > ut (l) = ut (r). Any voter type with t < 0 has
ut (l) > ut (r), while any voter with t > 0 has ut (l) < ut (r). However, it could be that some
of these types prefer ut (m) to either of the other two. In order to check this I calculate the
derivative of each of the expected utilities with respect to t.
D − 3 − 4sl
d[u(l)]
= −2t +
dt
D
d[u(m)]
= −2t
dt
d[u(r)]
D + 1 − 4sl
= −2t +
dt
D
s
1−δ
Q2
1 − δ D+1
2D
!
s
1−δ
Q2
1 − δ D+1
2D
!
When sl < D−3
then for any t < 0 we have d[u(m)]
< d[u(l)]
< d[u(r)]
. Utility is increasing for
4
dt
dt
dt
d[u(m)]
d[u(l)]
all three cases as we move towards 0. Given dt < dt for any t < 0 and ut (m) > ut (l) =
ut (r) for t = 0, it must be that for sl < D−3
there is no type with ut (l) > ut (m), ut (r). When
4
d[u(m)]
sl > D+1
then for any t > 0 we have dt < d[u(r)]
< d[u(l)]
. Combined, with the fact that
4
dt
dt
we have ut (m) > ut (l) = ut (r) for t = 0, this means that for sl > D+1
there is no type with
4
ut (r) > ut (m), ut (l).
What this means is that, conditional on sm = D−1
, if sl < D−3
, a district in which m and
2
4
r are the serious candidates will have no misaligned voting; and if sl > D+1
then a district
4
in which m and l are the serious candidates will have no misaligned voting.
However, each equilibrium has misaligned voting in a least one district. Recall that
E(S−dl ) = (sl − 1, sm , sr ), E(S−dm ) = (sl , sm − 1, sr ) and E(S−dr ) = (sl , sm , sr − 1). As
, dl and dr districts will have the same
all the relevant decisive events occur at sm = D−1
2
“route” to being decisive. That is, in any equilibrium if dl districts are conditioning on
(k, D−1
, D−1
− k), then dr districts must be conditioning on (k + 1, D−1
, D−1
− (k + 1)).
2
2
2
2
D−3
0
When 0 < sl < 4 , all dm and dr districts are conditioning on λ(2 ) events. In any of
49
these districts if the serious candidates are m and r, there is no misaligned voting. However,
we know that dl districts must either be conditioning on a λ(20 ) event or else a λ(3) event (if
it conditions on S−d = (0, D−1
, D−1
)). Whichever one of these is the case, there will always
2
2
, D−1
),
be misaligned voting in these dl districts. Indeed, if it conditions on S−d = (0, D−1
2
2
and the other districts are all races between m and r, it must be that there is only misaligned
< sl < D−1
case gives the same insight
voting in this single dl district. Examining the D+1
4
2
for the mirror case; there’ll be no misaligned voting in dl or dm districts if they focus on
races between l and m, but there will always be misaligned voting in the dr districts.
50
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